MGU Mathematics Series — Algebra I
Part III: Functions, Graphs, and Linear Models
Functions · Notation · Domain · Range · Representations
Chapter 5 of 14
MGU Mathematics Series  ·  Algebra I  ·  Library Textbook Edition

Chapter 5:
Introduction
to Functions

Relations, inputs and outputs, function notation, evaluation, tables, graphs, domain and range, discrete and continuous functions, and algebraic models — the foundational language of every function family that follows

■ Math Level: Algebra I ■

■ 14 Sections  •  Part III of 5 ■

Follows Chapter 4: Linear Inequalities  ·  Prepares students for Chapters 6, 7, and 8 — Graphing and Linear Models

Chapter 5 — Front Matter

Purpose and Overview

This chapter introduces one of the most important ideas in all of mathematics: the function. A function is a rule that assigns each input to exactly one output. That idea sounds simple, but it opens the door to nearly every topic that follows in algebra, geometry, statistics, science, finance, and computing.

Before this chapter, you worked with expressions that describe quantities and equations that describe conditions. Now you will begin studying relationships — situations where one quantity depends on another. Functions are the mathematical tool for describing, analyzing, and predicting those relationships.

By the end of this chapter, you will be able to read and write function notation, evaluate functions from equations, tables, and graphs, identify domains and ranges, and describe functions in words and symbols. This chapter prepares you for the deep work with linear functions in Chapters 6, 7, and 8, and for every function family you will encounter throughout Algebra I, Algebra II, and beyond.

Opening Question

A vending machine accepts a dollar bill and dispenses one item when you press a button. If you press button A3, you always get the same product. If you press button B7, you always get a different but consistent product. The machine never gives you two different items for the same button press.

In what ways is this vending machine behaving like a mathematical function? In what ways would it fail to be a function if it sometimes gave you two different items for the same button press? Keep this image in mind as you work through the chapter. A function is, at its core, a reliable rule: one input, one output, every time.

14Sections
10Objectives
16Key Terms
10+Worked Examples

Why This Chapter Matters

Before algebra, most of the mathematics you studied focused on computing specific numbers: the sum of two values, the area of a shape, the percent of a total. In algebra, the focus shifts toward relationships between quantities. How does the total cost change as the number of items increases? How does the height of a thrown ball change over time? How does a person's pay depend on the number of hours worked?

The concept of a function gives mathematicians and scientists a precise language for describing these relationships. Nearly every equation you write in Algebra I can be thought of as a function. Nearly every graph you draw represents a function. And nearly every real-world model in finance, science, and technology is built from functions.

Learning Objectives

  1. 1Define a function and explain what makes a relationship a function.
  2. 2Identify inputs, outputs, independent variables, and dependent variables.
  3. 3Read, write, and evaluate expressions using function notation.
  4. 4Build tables of values for a function from a rule.
  5. 5Identify whether a graph represents a function using the vertical line test.
  6. 6State the domain and range of a function from a table, graph, or equation.
  7. 7Distinguish between discrete and continuous functions.
  8. 8Write and interpret function rules from patterns and real-world situations.
  9. 9Compare multiple representations of the same function.
  10. 10Recognize and correct common mistakes involving functions.
Chapter 5 — Contents

Sections at a Glance

Sections 5.1 – 5.3
Relations, Inputs, Outputs, and Variables
What a Function Is · Inputs and Outputs · Independent and Dependent Variables
Section 5.1

What a Function Is

A Starting Point: Relations

Before defining a function precisely, it helps to think about a broader idea. A relation is any pairing of input values with output values. A relation simply associates some set of inputs with some set of outputs. There are no restrictions on how the pairing works.

Here is an example of a relation expressed as a set of ordered pairs:

{(1, 3), (2, 5), (3, 3), (4, 7)}

In this relation, the input 1 is paired with the output 3, the input 2 is paired with the output 5, and so on. Notice that the inputs 1 and 3 are both paired with the same output, 3. That is perfectly acceptable in a relation — different inputs can share the same output.

Now consider a different relation:

{(1, 3), (1, 5), (2, 6), (3, 4)}

Here, the input 1 is paired with two different outputs: both 3 and 5. This kind of ambiguity is the key distinction that separates a function from a general relation.

Definition — Function

A function is a relation in which every input is paired with exactly one output.

The input set is called the domain. The set of actual outputs is called the range.

Key Distinction — Inputs vs. Outputs

Repeated outputs are allowed. If input 3 always maps to output 7, and input 5 also maps to 7, the relation can still be a function. But if input 3 maps to both 7 and 9, that input is ambiguous and the relation is not a function. Focus only on repeated inputs when testing for functions.

Example 5.1.1
Identifying functions from ordered pairs
  • ProblemDetermine whether each relation is a function.

    (a) {(2, 4), (3, 9), (4, 16), (5, 25)}
    (b) {(1, 6), (2, 7), (1, 8), (3, 9)}
    (c) {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)}
  • ThinkFor each relation, list the input values. Check whether any input appears more than once with different outputs.
  • Solve (a) Inputs: 2, 3, 4, 5 — all different. → Function. (b) Input 1 appears twice: paired with 6 and with 8. → Not a function. (c) Inputs: 0, 1, 2, 3, 4 — all different. → Function.
  • CheckIn part (b), re-read (1, 6) and (1, 8). Both have the same input. If asked "What is the output for input 1?", there would be no single correct answer.
Answer: (a) Function.   (b) Not a function.   (c) Function.

Mapping Diagrams

Another way to represent a relation is a mapping diagram. A mapping diagram shows two sets — one for inputs and one for outputs — with arrows showing which inputs connect to which outputs. In a function, every input has exactly one outgoing arrow. No input can have two arrows leaving it.

Diagram Instruction — Mapping Diagrams

Draw two vertical ovals side by side. Label the left oval "Input" and the right oval "Output." Inside the left oval, list the values 1, 2, 3. Inside the right oval, list the values 5, 7, 9. Draw one arrow from 1 to 5, one arrow from 2 to 7, and one arrow from 3 to 9. This mapping represents a function because each input has exactly one outgoing arrow. Now draw a second mapping where the input 2 has arrows going to both 7 and 9 simultaneously — this represents a non-function because one input has two outgoing arrows.

Practice 5.1Concept Check
  1. In your own words, explain what makes a relation a function. Give one example of each.
  2. Does a function allow the same output to appear twice? Explain with an example.
  3. Consider the relation {(2, 5), (4, 5), (6, 5), (8, 5)}. Is this a function? Justify your answer.
  4. Consider the relation {(3, 7), (5, 9), (3, 11), (7, 13)}. Is this a function? Justify your answer.
Section 5.2

Inputs and Outputs

Every function involves two roles: the input and the output. The input is the value you start with — what you put into the function. The output is the result — what the function produces from that input.

SituationInputOutput
Total cost of buying applesNumber of poundsTotal price
Area of a squareSide lengthArea
Pay for hourly workHours workedTotal pay
Temperature conversionCelsius temperatureFahrenheit temperature
Projectile height over timeTime in secondsHeight in feet
Diagram Instruction — Function Machine

Draw a box labeled "Function Machine" in the center. On the left side, draw an arrow entering the box labeled "Input x." On the right side, draw an arrow leaving the box labeled "Output f(x)." Inside the box, write a rule such as "Multiply by 2, then add 3." The machine takes an input, applies the rule, and returns exactly one output. Same input always returns same output — that consistency is the defining feature of a function.

Practice 5.2Concept and Application
  1. For each situation, identify the input and the output.

    a. The number of calories burned depends on the number of minutes exercised.
    b. The cost of a taxi ride depends on the number of miles traveled.
    c. The amount of sales tax depends on the purchase price.
  2. Describe a real-world situation of your own that behaves like a function. Identify the input and output, and explain why each input produces exactly one output.
Section 5.3

Independent and Dependent Variables

VariableRoleDescription
Independent variableInputChosen freely; does not depend on the other variable
Dependent variableOutputDetermined by the input; its value depends on the independent variable
Tip — Which Variable Depends on Which?

Ask yourself: "Which quantity is being controlled or chosen, and which quantity is being produced?" The one being controlled is independent. The one being determined is dependent. In "total cost depends on hours worked," hours is independent and cost is dependent.

In most algebra problems, the independent variable is called x and the dependent variable is called y. When you write y = 2x + 3, the variable x is the independent variable (input), and y is the dependent variable (output). In real-world problems, letters are often chosen to reflect the quantities involved:

t for time, h for height: h = −16t² + 64t n for number of items, C for cost: C = 1.5n + 0.75 d for distance, s for speed, t for time: d = st
Practice 5.3Concept and Skill
  1. For each function, identify the independent variable and the dependent variable.

    a. A plumber charges $50/hr. Let h = hours, C = total charge: C = 50h.
    b. A car travels 60 mph. Let t = time, d = distance: d = 60t.
    c. A balloon rises 5 feet per second. Let s = seconds, H = height: H = 5s.
  2. Choose an appropriate variable name and define it clearly for each: (a) gallons of water in a tank, (b) price of a single apple, (c) days since a deposit was made.
Sections 5.4 – 5.7
Function Notation, Evaluation, Tables, and Graphs
Function Notation · Evaluating Functions · Tables of Values · Graphs and the Vertical Line Test
Section 5.4

Function Notation

When you write y = 2x + 3, the notation works, but it does not tell you the function's name, or how to clearly indicate evaluation at a specific input. Function notation was created to solve these problems.

Function Notation — Standard Form

f(x) = [rule]

Read: "f of x equals [rule]." Here, f is the function's name. The x inside the parentheses is the input variable. The rule describes what happens to the input.

Example: f(x) = 2x + 3 — the function named f multiplies the input by 2 and adds 3.

Critical Clarification — f(x) Is Not Multiplication

The notation f(x) does not mean f multiplied by x. The parentheses here are a notation convention meaning "the function f applied to the input x." This is one of the most common misreadings in Algebra I. f(3) does not mean f × 3; it means "evaluate the function at the input 3."

Naming Functions in Context

Functions are most commonly named f, g, or h, but any letter can be used. In applied problems, the function is often named after the quantity it produces:

Function NameInput VariableWhat It Represents
C(n)n = number of itemsCost function
h(t)t = time in secondsHeight function
A(s)s = side lengthArea function
P(h)h = hours workedPay function
Practice 5.4Reading Function Notation
  1. Rewrite each equation using function notation. Name the function f.

    a. y = 4x − 1    b. y = x² + 5    c. y = −3x + 7
  2. A student says "f(x) means f times x." Explain why this is incorrect. Give a correct interpretation.
  3. If C(n) = 2.50n + 10 represents the cost of ordering n items, what does C(6) ask for?
Section 5.5

Evaluating Functions

Evaluating a function means substituting a specific value for the input variable and computing the output. The process always follows the same steps: identify the input value → substitute using parentheses → simplify using the order of operations → state the output.

Example 5.5.1
Evaluating a function at a single value
  • ProblemLet f(x) = 3x − 5. Find f(2), f(−1), and f(0).
  • ThinkReplace x with the given value in each case. Simplify completely.
  • Solve f(2) = 3(2) − 5 = 6 − 5 = 1 f(−1) = 3(−1) − 5 = −3 − 5 = −8 f(0) = 3(0) − 5 = 0 − 5 = −5
  • CheckFor f(2): 3(2) − 5 = 6 − 5 = 1. Correct. ✓
Answer: f(2) = 1,   f(−1) = −8,   f(0) = −5.
Example 5.5.2
Evaluating a quadratic function with negative inputs
  • ProblemLet g(x) = x² − 4x + 1. Find g(3) and g(−2).
  • ThinkSubstitute carefully. When squaring a negative number, use parentheses around the entire substituted value first.
  • Solve g(3) = (3)² − 4(3) + 1 = 9 − 12 + 1 = −2 g(−2) = (−2)² − 4(−2) + 1 = 4 + 8 + 1 = 13
  • CheckFor g(3): 9 − 12 + 1 = −2. Correct. ✓
Answer: g(3) = −2,   g(−2) = 13.
Example 5.5.3
Evaluating with a variable expression as input
  • ProblemLet f(x) = 2x + 5. Find f(a) and f(x + 1).
  • ThinkThe input does not have to be a number. Replace every instance of x with the new input expression.
  • Solve f(a) = 2a + 5 f(x+1) = 2(x + 1) + 5 = 2x + 2 + 5 = 2x + 7
  • CheckTest with x = 0: f(1) = 2(1) + 5 = 7. From the simplified form: 2(0) + 7 = 7. These agree. ✓
Answer: f(a) = 2a + 5,   f(x + 1) = 2x + 7.
Student Warning — Parentheses for Negative Inputs

Always put parentheses around a negative substituted value. Writing f(−3) = 2 · −3 + 5 is sloppy and leads to sign errors. Write f(−3) = 2(−3) + 5 instead. This habit prevents the most common evaluation mistake in Algebra I.

Practice 5.5Evaluating Functions
  1. f(x) = 4x − 1. Find f(3), f(0), f(−2).
  2. g(x) = x² + 2. Find g(1), g(−3), g(0).
  3. h(x) = −3x + 7. Find h(2), h(−1), h(5).
  4. f(x) = x/2 + 1. Find f(4), f(−4), f(1).
  5. p(t) = 16t + 5. Find p(0), p(3), p(−2).
  6. f(x) = 2x + 3. Find f(a) and f(x − 1).
  7. g(x) = x² − 1. Find g(a + 1). Expand and simplify.
  8. g(x) = x² + x. Find g(3) and g(−3). Are the values equal?
Section 5.6

Tables of Values

A table of values is an organized way to record several input-output pairs for a function. Tables make the function's behavior visible across multiple inputs at once, allow you to look for patterns, and serve as a bridge between the rule and the graph.

Example 5.6.1
Building a table of values
  • ProblemBuild a table of values for f(x) = 2x + 1 using x = −2, −1, 0, 1, 2, 3.
  • ThinkSubstitute each input into the rule and compute the output.
  • Solve
    xf(x) = 2x + 1f(x)
    −22(−2) + 1−3
    −12(−1) + 1−1
    02(0) + 11
    12(1) + 13
    22(2) + 15
    32(3) + 17
  • CheckFor x = 2: 2(2) + 1 = 5. The table shows 5. ✓
Answer: Table as shown above. Notice the output increases by 2 each time x increases by 1.
Reading Patterns from Tables

The output column above reads: −3, −1, 1, 3, 5, 7. Each time the input increases by 1, the output increases by 2. This constant increase reflects the structure of f(x) = 2x + 1 — the coefficient 2 controls how fast the output grows. You will study this idea in depth in Chapters 6 and 7 as slope and rate of change.

Example 5.6.2
Identifying a function from a table
  • ProblemDoes the following table represent a function?

    Input12334
    Output58111417
  • ThinkCheck whether any input value appears more than once with different outputs.
  • SolveInput value 3 appears twice, paired with outputs 11 and 14. Same input, two different outputs.
  • CheckIf asked "What is the output for input 3?", there is no single answer. That ambiguity confirms it is not a function.
Answer: Not a function — input 3 maps to two different outputs.
Practice 5.6Building and Reading Tables
  1. Complete a table of values for f(x) = 5x − 2 using x = −2, −1, 0, 1, 2.
  2. Complete a table of values for g(x) = x² − 3 using x = −2, −1, 0, 1, 2.
  3. Complete a table of values for h(x) = ½x + 3 using x = −4, −2, 0, 2, 4.
  4. State whether each table represents a function. Justify your answer.

    a. Inputs: 2, 4, 6, 8 — Outputs: 5, 9, 13, 17
    b. Inputs: 1, 2, 2, 3 — Outputs: 4, 7, 9, 12
    c. Inputs: 0, 1, 2, 3 — Outputs: 6, 6, 6, 6
Section 5.7

Graphs of Functions

The graph of a function is a visual picture of all its input-output pairs. Each ordered pair (x, f(x)) becomes a point in the coordinate plane. When all those points are plotted, they form the shape of the function — a line, a curve, a scatter of dots, or some other pattern.

The Vertical Line Test

The vertical line test formalizes the definition on a graph: if any vertical line crosses a graph in more than one point, then the graph does not represent a function. If every vertical line crosses the graph in at most one point, then the graph represents a function.

The test works because a vertical line represents a single input value. If the line hits the graph in two places, that input has two outputs — which violates the definition.

Diagram Instruction — Vertical Line Test

Draw two graphs side by side. On the left, draw a straight diagonal line across the coordinate plane. Then draw several vertical dashed lines, each crossing the diagonal line at exactly one point. Label this "Passes the Vertical Line Test — Is a Function." On the right, draw a circle. Then draw one vertical dashed line cutting through the circle at two points (one on the top arc and one on the bottom arc). Label this "Fails the Vertical Line Test — Not a Function."

Example 5.7.1
Applying the vertical line test
  • ProblemDetermine whether each graph represents a function.

    (a) A straight line crossing the plane diagonally from lower-left to upper-right.
    (b) A circle centered at the origin.
    (c) A U-shaped parabola opening upward, lowest point at the origin.
  • ThinkFor each graph, imagine drawing vertical lines at different x-values. Would any vertical line hit the graph more than once?
  • Solve (a) Diagonal line: every vertical line crosses it exactly once. → Function. (b) Circle: a vertical line through the interior crosses it twice. → Not a function. (c) Parabola opening upward: every vertical line crosses it at most once. → Function.
Answer: (a) Function.   (b) Not a function.   (c) Function.
Diagram Instruction — Graphing from a Table

Using the table from Example 5.6.1, plot the ordered pairs (−2, −3), (−1, −1), (0, 1), (1, 3), (2, 5), (3, 7) in the coordinate plane. Draw a straight line through all six points. Label the line f(x) = 2x + 1. Notice that the points are collinear — they all lie on the same straight line — which reflects the constant rate of change built into the rule.

Practice 5.7Graphs and the Vertical Line Test
  1. A graph is a single straight horizontal line at y = 4. Does it pass the vertical line test? Is it a function? What is the range?
  2. A graph is a vertical line at x = 3. Does it pass the vertical line test? Is it a function? Explain.
  3. Plot the ordered pairs from Practice 5.6, problem 1 (f(x) = 5x − 2) on a coordinate plane. Connect the points. Describe the graph.
  4. For each description, state whether the graph could represent a function: (a) a parabola opening downward, (b) two separate lines that cross, (c) an S-shaped curve that passes through each x exactly once.
Sections 5.8 – 5.10
Domain, Range, Discrete and Continuous Functions, and Rules from Patterns
Domain & Range · Discrete vs. Continuous · Finding Rules from Tables
Section 5.8

Domain and Range

Two Key Descriptors of Every Function

Domain — the set of all allowable input values; all x-values for which the function is defined. Think of the domain as everything you are allowed to put into the function.

Range — the set of all output values the function actually produces when applied to its domain. Think of the range as everything that comes out.

Domain and Range from Tables and Graphs

When a function is given as a finite table of ordered pairs, the domain is the list of all input values and the range is the list of all output values. When a function is given as a graph, the domain is the set of all x-values the graph covers (reading left to right), and the range is the set of all y-values the graph covers (reading bottom to top).

Domain Restrictions

Some functions have natural domain restrictions — values that are not allowed because they would make the output undefined or impossible. The most common situations in Algebra I are:

SituationRestrictionExample
Division by zeroExclude values that make the denominator zerof(x) = 1/x; x ≠ 0
Square root of a negativeExclude inputs that produce a negative radicandf(x) = √x; x ≥ 0
Real-world contextExclude values that are physically impossibleHours worked ≥ 0; items ordered ≥ 0
Example 5.8.1
Identifying domain and range
  • ProblemState the domain and range of each function.

    (a) {(0, 1), (1, 4), (2, 9), (3, 16)}
    (b) f(x) = x + 5, with no context restrictions.
    (c) A graph: horizontal line segment from (−3, 4) to (5, 4), including both endpoints.
  • ThinkFor ordered pairs, list inputs and outputs. For an equation with no restrictions, domain and range are all real numbers. For a graph, read horizontal span for domain and vertical span for range.
  • Solve (a) Domain: {0, 1, 2, 3} Range: {1, 4, 9, 16} (b) Domain: all real numbers (no inputs make the rule undefined) Range: all real numbers (x + 5 takes every real value) (c) Domain: −3 ≤ x ≤ 5 (horizontal span from −3 to 5) Range: {4} (graph stays at y = 4 throughout)
Answer: (a) D: {0,1,2,3}, R: {1,4,9,16}.   (b) D: all reals, R: all reals.   (c) D: [−3, 5], R: {4}.
Memory Aid — Domain Before Range

D comes before R in the alphabet. Inputs come before outputs in the function process. Domain is inputs; Range is outputs. The domain and range are not always the same — pay attention to what the output actually does, not just what the inputs are.

Practice 5.8Domain and Range
  1. State the domain and range of each function.

    a. {(1, 5), (2, 8), (3, 11), (4, 14)}
    b. {(0, 0), (1, 1), (4, 2), (9, 3)}
    c. f(x) = 3x + 1 with domain {0, 1, 2, 3}
    d. A function whose graph is a line extending infinitely in both horizontal directions.
  2. A function describes the number of days in month m of a non-leap year, where m = 1 is January and m = 12 is December. State the domain and identify the possible range values.
Section 5.9

Discrete and Continuous Relationships

TypeDomainReal-World ExamplesGraph Appearance
DiscreteIsolated, separate values (often whole numbers)Number of tickets sold, items ordered, students in a classSeparate dots — never connected
ContinuousUnbroken interval of valuesTime, temperature, distance, weightConnected curve or line
Tip — How to Decide

Ask: can the input take any value within a range, including fractions and decimals? If yes, it is likely continuous. If the input can only take specific isolated values (typically whole numbers), it is likely discrete. Selling 2.7 tickets makes no sense — discrete. A balloon at height 4.37 feet at time 1.85 seconds makes perfect sense — continuous.

Example 5.9.1
Identifying discrete and continuous functions
  • ProblemDecide whether each situation is discrete or continuous.

    (a) Cost of buying whole boxes of crackers at $3 each, as a function of the number of boxes.
    (b) Height of a rising balloon as a function of time in seconds.
    (c) {(1, 3), (2, 6), (3, 9), (4, 12)}.
  • ThinkCan the input take any value in a range (continuous), or only specific isolated values (discrete)?
  • Solve (a) Only whole numbers of boxes: {0, 1, 2, 3, ...} → Discrete. (b) Time flows continuously; height exists at every fraction of a second → Continuous. (c) Only four specific input values given → Discrete.
Answer: (a) Discrete.   (b) Continuous.   (c) Discrete.
Student Warning — Do Not Connect Dots on a Discrete Graph

If a function is discrete, plot points only. Do not connect them with a line or curve. Connecting the points suggests the function is defined between them — which it is not. Graphing the number of books checked out per day calls for separate points, not a connected curve.

Practice 5.9Discrete and Continuous
  1. Classify each as discrete or continuous. Justify your answer.

    a. The temperature of water as it heats from 20°C to 100°C.
    b. The number of students who pass a test each semester.
    c. A car's speed at any given moment during a trip.
    d. The number of goals scored in a soccer game.
  2. A function f is given by the table: inputs 1, 2, 3, 4; outputs 10, 20, 30, 40. Sketch how you would graph this function. Should you connect the points? Why or why not?
Section 5.10

Function Rules from Patterns

A table of values shows the behavior of a function, but not always the rule. For linear functions, the output changes by a constant amount each time the input increases by 1. When you find that constant change, you can identify the rule.

Example 5.10.1
Finding a linear rule from a table
  • ProblemFind the function rule for this table.

    x01234
    f(x)47101316
  • ThinkLook at how the output changes as x increases by 1. Then identify the starting value when x = 0.
  • Solve Change from x=0 to x=1: 7 − 4 = 3 Change from x=1 to x=2: 10 − 7 = 3 Change is constant: +3 each time → coefficient of x is 3. When x = 0, f(x) = 4 → constant term is 4. Rule: f(x) = 3x + 4
  • CheckFor x = 3: f(3) = 3(3) + 4 = 9 + 4 = 13. Table shows 13. ✓
Answer: f(x) = 3x + 4.
Connecting to Real Situations

A gym charges a one-time sign-up fee of $20 and then $15 per month. The total cost C after m months is C(m) = 15m + 20. The rule comes directly from the verbal description: the recurring monthly charge is the coefficient, and the one-time fee is the constant. This pattern — constant rate plus starting value — defines linear functions, which are the focus of Chapters 6 and 7.

Practice 5.10Rules from Patterns
  1. Find the function rule for each table.

    a. x: 0, 1, 2, 3 — f(x): 2, 5, 8, 11
    b. x: 0, 1, 2, 3 — f(x): 10, 7, 4, 1
    c. x: 0, 2, 4, 6 — f(x): 3, 7, 11, 15
  2. A taxi charges a $3 base fee plus $2.50 per mile. Write the function rule C(m) for the total cost, where m is miles traveled.
Sections 5.11 – 5.14
Comparing Representations, Real Situations, Mistakes, and Looking Ahead
Four Representations · Functions in Context · Common Mistakes · Preparing for Chapter 6
Section 5.11

Comparing Representations

A single function can be expressed in four main ways. Understanding how to move between representations is a core algebra skill that you will use throughout the entire course.

RepresentationStrengthsLimitations
Verbal descriptionEasy to communicate; connects to contextImprecise; hard to compute from
Equation or ruleExact; compact; good for computingAbstract; less visual
Table of valuesConcrete; easy to read specific valuesShows only selected inputs; not full behavior
GraphVisual; shows overall behavior clearlyApproximate for non-integer values
Example 5.11.1
Moving between all four representations
  • ProblemA plumber charges a flat fee of $50 for a service visit, plus $30 per hour of work. Provide all four representations.
  • VerbalThe total cost equals 30 times the number of hours worked, plus the flat fee of 50 dollars.
  • Equation C(h) = 30h + 50
  • Table
    h (hours)01234
    C(h) (dollars)5080110140170
  • GraphThe graph begins at (0, 50) and rises steadily to the right. For every 1-unit increase in hours, the cost increases by $30. The graph is a straight line with positive slope, starting at the point (0, 50) on the vertical axis.
  • CheckFor h = 2: C(2) = 30(2) + 50 = 60 + 50 = 110. The table shows 110. ✓
Answer: C(h) = 30h + 50. All four representations describe the same function.
Same Function, Four Languages

The verbal description, equation, table, and graph in the example above all describe the same function. They are four different languages for one mathematical object. One of the great skills of algebra is the ability to read any of these representations and translate freely between them. This skill grows with every chapter you study.

Practice 5.11Comparing Representations
  1. A parking garage charges $2.50 per hour. Let h = hours and C = total cost. Provide: (a) the verbal description of the rule, (b) the function equation, (c) a table for h = 0, 1, 2, 3, 4, and (d) a description of what the graph would look like.
  2. A function rule is f(x) = 4x − 3. Build a table for x = 0, 1, 2, 3, 4. Then describe what the corresponding graph would look like, including the starting point and direction.
  3. A pool is being drained at 200 gallons per hour from an initial 1,000 gallons. Let t = hours and V = volume. Write the function V(t) and build a table for t = 0, 1, 2, 3, 4. At what time does the pool empty?
Section 5.12

Functions in Real Situations

Functions allow you to take a real situation, strip it down to its mathematical structure, and use that structure to compute, predict, and explain. Here are four examples from contexts you will encounter throughout the MGU Mathematics Series.

Temperature Conversion

The function that converts Celsius temperatures to Fahrenheit is:

F(C) = (9/5)C + 32

The input is a Celsius temperature C, and the output is the equivalent Fahrenheit temperature. Evaluating: F(0) = 32 (freezing point), F(100) = 212 (boiling point).

Hourly Wages

If a worker earns $14.50 per hour with no overtime:

P(h) = 14.5h (h = hours worked, h ≥ 0)

The domain is restricted to non-negative values. The range is also non-negative.

Area as a Function of Side Length

The area of a square with side length s:

A(s) = s² (s > 0)

This is a quadratic function — a function family you will study in depth in Chapter 12.

Simple Interest Savings

A savings account earning simple interest of 4% per year on a $1,000 deposit:

B(t) = 1000(1 + 0.04t) (t = years)

The input is time, the output is the account balance. This is a linear function of t. Compound interest, studied in Algebra II, produces an exponential function.

Practice 5.12Real-World Applications
  1. Temperature Conversion. Using F(C) = (9/5)C + 32:
    a. Find the Fahrenheit equivalent of 25°C and −10°C.
    b. Find the Celsius temperature at which F = 98.6°F.
  2. Hourly Wages. A part-time employee earns $12.75/hr.
    a. Write W(h) for weekly earnings. Find W(15), W(20), W(40).
    b. State the domain in a real-world context. Is the function discrete or continuous?
  3. Phone Plan. A plan costs $25 per month plus $0.10 per text message.
    a. Write C(n) for monthly cost. Build a table for n = 0, 50, 100, 200.
    b. How many messages can be sent on a $35 budget?
  4. Falling Object. The height of a dropped ball (feet) is h(t) = 100 − 16t², dropped from 100 feet.
    a. Find h(0), h(1), h(2).
    b. Describe what the decreasing values mean in context.
    c. Is this function discrete or continuous? Why?
Section 5.13

Common Function Mistakes

Mistake 1 — Misreading f(x) as Multiplication
f(4) means f × 4, so if f(4) = 14 then f = 3.5

The parentheses in f(x) mean "applied to," not "multiplied by." f(4) means "evaluate the function at input 4." It does not mean f times 4.

Mistake 2 — Confusing Domain and Range
The range of a function is the set of all input values.

Domain is always inputs (independent variable). Range is always outputs (dependent variable). D comes before R; inputs come before outputs.

Mistake 3 — Repeated Outputs Disqualify a Function
The table {(1,3), (2,7), (3,3), (4,11)} is not a function because output 3 appears twice.

Repeated outputs are allowed. What disqualifies a function is the same input giving two different outputs. In the table above, each input is unique, so it is a valid function.

Mistake 4 — Omitting Parentheses on Negative Inputs
g(−3) = −3² − 5(−3) = −9 + 15 = 6

Without parentheses, −3² evaluates as −(3²) = −9. The correct approach: g(−3) = (−3)² − 5(−3) = 9 + 15 = 24. Always substitute negative values inside parentheses.

Mistake 5 — Connecting Dots on a Discrete Graph
A student graphs the number of siblings per student by connecting the plotted points with a smooth curve.

A discrete function's graph should show individual points only. Connecting them claims the function is defined between the dots — which it is not. You cannot have 2.4 siblings.

Mistake 6 — Not Recognizing a Constant Function
f(x) = 5 is not a real function because x does not appear in the rule.

A constant function is a valid function. Every input maps to exactly one output — always 5 in this case. The graph is a horizontal line. Every input has exactly one (constant) output.

Practice 5.13Error Analysis
  1. A student is given f(x) = 3x + 2 and evaluates f(4) = 14. The student then writes: "So f × 4 = 14, which means f = 3.5." What error was made in the second step?
  2. A student claims the table {(1,3), (2,7), (3,3), (4,11)} does not represent a function because output 3 appears twice. Is the student correct? Explain.
  3. Let g(x) = x² − 5x. A student finds g(−3) = −3² − 5(−3) = −9 + 15 = 6. Identify and correct the error.
  4. A student graphs the number of concert tickets sold each week for 5 weeks by connecting the five plotted points with a smooth curve. Why is this a mistake?
  5. A student says f(x) = 7 cannot be a function because there is no rule. Explain what f(x) = 7 actually means and why it is a valid function.
Section 5.14

Preparing for Linear Functions

In Chapter 6, you will deepen your work with the coordinate plane, slope, and graphs. In Chapter 7, you will study the linear function family in full — slope-intercept form, point-slope form, standard form, and all the connections between equations, graphs, tables, and verbal descriptions. Everything you have built in this chapter is the foundation for that work.

As you move into Chapter 6, keep asking the questions this chapter has trained you to ask: What is the input? What is the output? What is the rule? What does the graph look like? What does the domain allow? What does the range tell me about the function's behavior? These questions work for every function family you will encounter — not just linear ones.

Practice 5.14Readiness Check
  1. Evaluate f(x) = 2x + 5 for x = 6. Then find the value of x that makes the expression equal to 17.
  2. Evaluate 3x − 4 for x = 3, 4, and 5. Which value makes the expression equal to 8?
  3. A rental car costs $40/day plus a $15 fee. Write a function C(d) for total cost. For how many days can you rent for $175?
  4. Two students simplify f(x) = 3(x + 2) − x. Student A gets f(x) = 2x + 6. Student B gets f(x) = 2x + 2. Which student is correct? Show all steps.
  5. A function has the table: x = 1, 2, 3, 4, 5 and f(x) = 4, 7, 10, 13, 16. Find the rule, evaluate f(10), and describe what the graph would look like.
Chapter 5 — Back Matter

Chapter Summary

Key Terms Review

function
A relation in which each input is assigned exactly one output. Same input always produces same output.
input
The value put into a function; the independent variable; the x-value in most algebra contexts.
output
The value produced by a function; the dependent variable; the result of applying the rule to an input.
relation
Any pairing of input values with output values, with no restriction on how inputs and outputs are matched.
mapping
A diagram showing two sets (inputs and outputs) with arrows connecting each input to its output(s).
independent variable
The variable representing the input; chosen freely; does not depend on the other variable in the relationship.
dependent variable
The variable representing the output; its value is determined by the independent variable.
function notation
The form f(x) used to name a function and denote its input. Not multiplication — means "function f applied to input x."
evaluate
Substitute a specific input value into a function rule and simplify to find the output.
table of values
An organized list of input-output pairs for a function; used to identify patterns and build graphs.
vertical line test
A visual test: if any vertical line crosses a graph more than once, the graph does not represent a function.
domain
The set of all allowable input values for a function. Think: everything you can put in.
range
The set of all actual output values produced by a function. Think: everything that comes out.
discrete function
A function with a domain of isolated, separate values. Graphed as individual points — never connected.
continuous function
A function defined over an unbroken interval. Graphed as a connected curve or line.
function rule
The algebraic expression describing how inputs are mapped to outputs. For example: f(x) = 3x + 4.

Concept Review Questions

Answer in complete sentences.

  1. What is the difference between a relation and a function?
  2. Explain what it means for a function to assign "exactly one output" to each input. Why is this requirement important?
  3. In a mapping diagram, how can you tell whether the diagram represents a function?
  4. What is the difference between the independent variable and the dependent variable in a function?
  5. Explain the function notation f(x) to someone who has never seen it. What does the letter f represent? What does x represent?
  6. A student says f(3) means f times 3. Explain the error and give the correct interpretation.
  7. What is the vertical line test, and why does it work?
  8. What is the domain of a function? What is the range?
  9. Give an example of a discrete function situation and a continuous function situation. Explain the difference.
  10. What are the four representations of a function? Describe the strength of each.

Skill Practice

Part AFunction Identification

Determine whether each relation is a function. Justify your answer.

  1. {(2,5), (3,8), (4,11), (5,14)}
  2. {(1,3), (2,6), (1,9), (3,12)}
  3. Inputs {A,B,C} all map to output {10}
  4. Inputs {1,2,3} map to outputs {4,4,4}
  5. {(0,0),(1,1),(−1,1),(2,4),(−2,4)}
  6. {(5,1),(5,2),(5,3)}
Part BEvaluating Functions
  1. f(x) = 5x − 3: Find f(0), f(2), f(−4).
  2. g(x) = x² + 2x − 1: Find g(3), g(−2).
  3. h(x) = −4x + 20: Find h(0), h(5), h(−1).
  4. p(t) = 16t²: Find p(0), p(1), p(3).
  5. f(x) = 3x + 4: Find f(x + 2). Expand and simplify.
  6. g(x) = x² − x: Find g(−4). Show parentheses step.
Part CTables and Domain/Range
  1. Complete a table for f(x) = 2x² − 1 using x = −2, −1, 0, 1, 2.
  2. Find the rule for: x = 0,1,2,3 — f(x) = 1,4,7,10.
  3. Find the rule for: x = 0,1,2,3 — f(x) = 8,5,2,−1.
  4. State domain and range: {(−3,2), (0,5), (4,2), (7,−1)}.
  5. State domain and range: f(x) = 2x + 7 with domain {−1, 0, 1, 2}.
  6. Is the table a function? Inputs: 3,5,7,9 — Outputs: 2,4,2,6.
Part DDiscrete and Continuous; Representations
  1. Classify: temperature rising as a cake bakes.
  2. Classify: number of goals scored in each game.
  3. Three isolated points: (1,4), (3,8), (5,12). Write the rule. Is this discrete or continuous?
  4. Write C(m) for a cell plan: $30/month + $0.05/min. Find C(0), C(100), C(400).
  5. A function's graph is a line starting at (0,3) and rising to (5,13). State the domain, range, and rule.
  6. A musician earns $75 per performance. Write E(p). Is this discrete or continuous?

Word Problems

For each problem: define your variable, write a function rule, and answer the specific question.

  1. A gym membership costs $50 per month. Write a function T(m) for total cost after m months. How much after one full year?
  2. A store sells pens for $1.25 and notebooks for $3.00. Write a function for the total cost of p pens and n notebooks. Find the total for 4 pens and 3 notebooks.
  3. A car travels at 55 mph. Write D(t) for distance in t hours. How far in 3.5 hours?
  4. A student scored 72 on the first test and p additional points on the second. Write a function for the combined score. If the combined score is 155, how many points on the second test?
  5. A field has length 3x meters and width (x + 5) meters. Write a simplified function for the perimeter in terms of x.
  6. A business sells candles for $8 each. Materials cost $3 per candle; monthly overhead is $200. Write a function P(c) for monthly profit on c candles. Find profit for 60 candles.

Error Analysis

Each problem shows a student's work with an error. Identify the error and write the correct solution.

Problem 1
f(x) = 3x + 2. Student finds f(4) = 14. Then writes: f·4 = 14, so f = 3.5.

What error was made in the second step? What does f(4) actually mean?

Problem 2
g(−3) = −3² − 5(−3) = −9 + 15 = 6 where g(x) = x² − 5x

Identify the error and show the correct computation.

Problem 3
A student connects five isolated points (tickets sold each week) with a smooth curve.

Why is connecting the dots a mistake? How should this graph be drawn?

Problem 4
A student says the domain of f(x) = 2x + 7 is "the set of outputs."

Correct the statement. What is the domain? What is the range?

Problem 5
A student says {(1,3),(2,7),(3,3),(4,11)} is not a function because output 3 appears twice.

Is the student correct? Justify your answer with the definition of a function.

Chapter 5 Checkpoint

Chapter 5 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 5
Part A — Vocabulary and Concepts
  1. Write the definition of a function in your own words.
  2. What does the notation h(5) mean if h is a function?
  3. What is the domain of a function? What is the range?
  4. Describe the vertical line test. What does it determine?
  5. What is the difference between a discrete and a continuous function?
Part B — Function Identification
  1. Determine whether each relation is a function. Justify each answer.
    a. {(2,5), (3,8), (4,11), (5,14)}
    b. {(1,3), (2,6), (1,9), (3,12)}
    c. A mapping in which inputs {A, B, C} all map to the output 10.
Part C — Evaluating Functions
  1. Let f(x) = 5x − 3. Find: f(0), f(2), f(−4).
  2. Let g(x) = x² + 2x − 1. Find: g(3), g(−2).
  3. Let p(t) = −4t + 20. Find the value of t for which p(t) = 0.
Part D — Tables and Domain/Range
  1. Build a complete table of values for f(x) = 2x² − 1 using x = −2, −1, 0, 1, 2.
  2. State the domain and range of the function: {(−3, 2), (0, 5), (4, 2), (7, −1)}.
  3. Three isolated points: (1, 4), (3, 8), (5, 12). Is there a pattern? Write the function rule if one exists. Is this function discrete or continuous?
Part E — Real-World Application
  1. A musician earns $75 per performance.
    a. Write a function E(p) for total earnings where p is the number of performances.
    b. Find E(4) and E(12).
    c. State the domain and range in context.
    d. Is this function discrete or continuous? Explain.
  2. A taxi service charges $3.00 at the start of each ride plus $2.50 per mile.
    a. Write a function C(m) for the cost of a ride of m miles.
    b. Build a table for m = 0, 1, 2, 5, 10.
    c. How much does a 7-mile ride cost?
Part F — Error Analysis
  1. A student evaluates f(−3) for f(x) = 4x − 1 and writes: "f(−3) = 4 · −3 − 1 = −13." The numerical answer is correct, but the work has a notation problem. Rewrite the work using proper parentheses. Then verify the answer.
  2. A student evaluates g(x) = x² − 5 at x = −3 and gets g(−3) = −9 − 5 = −14. Find and explain the error. What is the correct answer?
Looking Ahead
Chapter 6: The Coordinate Plane and Graphing Relationships

Chapter 5 established the function concept: input, output, rule, notation, domain, range, and the four representations. Chapter 6 now deepens your work with the visual representation — the coordinate plane. You will work with coordinates, axes, quadrants, intercepts, and the idea of slope as change. You will graph relationships from tables and verbal descriptions, and you will begin to see how the steepness of a line connects to the rate at which a function's output changes. Chapter 7 will then bring everything together into a full treatment of linear functions — the most important and widely used function family in all of Algebra I.

MGU Library Connection
  • Function Notation Reference — Appendix G of this textbook
  • Algebra Symbol Reference — Appendix D of this textbook
  • Algebra II, Chapter 1: Algebra II Readiness and Function Review (domain, range, composition, inverses, transformations)
  • Geometry: Coordinate geometry sections use function input-output thinking directly
  • Statistics: Regression and data modeling are built on the function concept
  • Physics: Kinematic equations (d = rt, h = −16t² + v₀t) are functions of time
  • Computer Science: Function design, input-output models, and parameter passing mirror the mathematical definition
  • MGU Dictionary entries: function, domain, range, independent variable, dependent variable, function notation, vertical line test