Chapter 1 — Front Matter

Purpose and Overview

This chapter is not a repetition of Algebra I. It is a bridge. If you completed Algebra I, you already know how to work with variables, write and evaluate expressions, solve equations and inequalities, use function notation, plot points, interpret graphs, and think about linear and quadratic relationships. Those ideas were not a destination. They were preparation.

Algebra II looks at familiar tools from a higher elevation. In Algebra I, you studied a few specific function families. In Algebra II, you will study many: polynomial, rational, radical, exponential, logarithmic, trigonometric, and more. This chapter reconnects you with the vocabulary, notation, and reasoning skills you will need every day in Algebra II — and extends each idea toward the depth the course demands.

Opening Question

A scientist monitors two bacterial populations. The first doubles every hour. The second grows by 500 organisms per hour. After one hour, both are equal in size. Which will be larger after ten hours? After twenty? How much larger? You may not be able to answer this yet — but by the end of this chapter you will have the function tools to set up the comparison, and by Chapter 9, the exponential machinery to answer it precisely.

14Sections

10Parent Functions

6Function Families

19Key Terms

Learning Objectives

1Describe what Algebra II studies and how it extends Algebra I.
2Evaluate and simplify algebraic expressions, including with expression inputs.
3Solve linear and quadratic equations and inequalities fluently.
4Use function notation fluently, including evaluation at expressions.
5Identify the domain and range of a function from tables, graphs, equations, and context.
6Read and interpret function graphs, including intercepts and zeros.
7Calculate average rate of change over an interval and interpret it in context.
8Apply vertical and horizontal transformations to function graphs.
9Identify and describe the ten major parent functions.
10Compute the composition of two functions and evaluate at specific inputs.
11Determine whether two functions are inverses; find inverses algebraically.
12Choose and build function models for real-world situations.
13Explain how function families studied in Algebra II are connected and distinct.

Chapter 1 — Contents

Sections at a Glance

1.1What Algebra II StudiesFunction families; polynomial, rational, radical, exponential, logarithmic, trigonometric; the organizing idea of the course
1.2Reviewing Algebraic ExpressionsTerms, coefficients, like terms; evaluating at values and expressions; distributive property; simplification
1.3Reviewing Equations and InequalitiesInverse operations; factoring; quadratic formula; inequality direction-reversal rule; solution sets as intervals
1.4Function Notation RevisitedReading f(x); evaluation at numbers and expressions; multiple function names; placeholder principle
1.5Domain and RangeAlgebraic restrictions; denominator and radical rules; contextual domain; range from graphs; interval notation
1.6Graphs as Function RepresentationsVertical line test; reading domain, range, intercepts, zeros, increase/decrease, end behavior from graphs
1.7Intercepts, Zeros, and SolutionsDefinitions and connections; y-intercept from f(0); x-intercepts from f(x) = 0; three names, one concept
1.8Average Rate of ChangeFormula and secant line interpretation; evaluating and interpreting in context; bridge to Calculus
1.9Transformations of FunctionsVertical and horizontal shifts; stretches and compressions; reflections; general form g(x) = a·f(b(x−h))+k
1.10Parent FunctionsTen parent functions; domain, range, and shape for each; reference for the entire Algebra II course
1.11Composition of Functions(f ∘ g)(x) = f(g(x)); evaluating at points and expressions; non-commutativity; uses in later chapters
1.12Inverse RelationshipsVerifying inverses by composition; finding inverses algebraically; graphs across y = x; horizontal line test
1.13Modeling with FunctionsChoosing function type; linear, quadratic, exponential, rational models; real-world interpretation
1.14Preparing for Advanced Function FamiliesHow each Chapter 1 concept reappears; preview of Algebra II arc; readiness summary

Section 1.1

What Algebra II Studies

Algebra I gave you the foundational tools of symbolic mathematics: variables, expressions, equations, and your first two function families — linear and quadratic. Those tools were preparation. Algebra II is the course where you study function families in depth. A function family is a group of related functions sharing the same general structure. In Algebra II, you add many more families to your collection — and learn that the same analytical tools apply across all of them.

Function Family	General Form	What It Describes
Polynomial	`aₙxⁿ + … + a₁x + a₀`	Generalizes linear and quadratic patterns; end behavior, zeros, turning points
Rational	`p(x) / q(x)`	Ratios of polynomials; restrictions, asymptotes, holes
Radical	`f(x) = ⁿ√x`	Roots, inverse power relationships, domain restrictions
Exponential	`f(x) = a · bˣ`	Multiplicative growth and decay; finance, science, population
Logarithmic	`f(x) = log_b(x)`	Answers exponent questions; inverse of exponential
Trigonometric	`sin(x), cos(x), tan(x)`	Triangle ratios and periodic cycles; waves, rotation

Student Note

Read carefully even when the early sections feel familiar. In Algebra II, revisiting a concept from Algebra I always means uncovering more structure — not repeating the same idea at the same level.

Section 1.2

Reviewing Algebraic Expressions

An algebraic expression is a mathematical phrase built from numbers, variables, and operations — no equals sign, no inequality. It represents a quantity. Every expression is built from terms: products of coefficients and variables raised to powers, or constants. Terms sharing the same variable part are like terms and can be combined by adding or subtracting their coefficients.

Example 1.2.1

Evaluating an expression at a negative value

ProblemEvaluate 2x² − 3x + 1 when x = −4.
ThinkReplace every x with (−4). Be careful squaring a negative value.
Solve2(−4)² − 3(−4) + 1 = 2(16) + 12 + 1 = 32 + 12 + 1 = 45
CheckThe squared term contributes 2 × 16 = 32. Adding 12 and 1 gives 45. ✓

Answer: 45

Example 1.2.2

Simplifying an expression with the distributive property

ProblemSimplify 3(2x − 4) − 5(x + 1).
ThinkDistribute each coefficient, then combine like terms.
Solve3(2x − 4) − 5(x + 1) = 6x − 12 − 5x − 5 = (6x − 5x) + (−12 − 5) = x − 17
CheckTest x = 2. Original: 3(0) − 5(3) = −15. Simplified: 2 − 17 = −15. ✓

Answer: x − 17

Section 1.3

Reviewing Equations and Inequalities

An equation states that two expressions are equal. Solving means finding every value that makes the statement true. An inequality is similar but uses a comparison symbol. Its solution is usually an interval of values rather than a single answer.

Example 1.3.1

Solving a linear equation

ProblemSolve 3(x − 2) + 4 = 2x − 1.
Solve3x − 6 + 4 = 2x − 1 3x − 2 = 2x − 1 x = 1
Checkx = 1: Left side = 3(−1) + 4 = 1. Right side = 2(1) − 1 = 1. ✓

Answer: x = 1

Example 1.3.2

Solving a quadratic equation by factoring

ProblemSolve x² − 5x + 6 = 0.
Solve(x − 2)(x − 3) = 0 x = 2 or x = 3
Checkf(2) = 4 − 10 + 6 = 0 ✓ | f(3) = 9 − 15 + 6 = 0 ✓

Answer: x = 2 or x = 3

Example 1.3.3

Solving a two-step inequality

ProblemSolve −3x + 5 > 11.
Solve−3x > 6 x < −2 ← direction reverses: dividing by −3
CheckTest x = −4: −3(−4) + 5 = 17 > 11 ✓. Test x = 0: 5 ≯ 11 — correctly excluded. ✓

Answer: x < −2 (open circle at −2, shaded left)

Student Warning

The direction-reversal rule for inequalities is one of the most frequently forgotten rules in algebra. Every time you multiply or divide an inequality by a negative number, reverse the symbol.

Section 1.4

Function Notation Revisited

A function is a rule that assigns exactly one output to each input. The notation f(x) is read "f of x" — the output of function f when the input is x. In Algebra II, functions are the primary objects of study, and you will work with function notation constantly.

Example 1.4.1

Evaluating a function at multiple inputs

ProblemLet f(x) = x² − 2x + 5. Find f(0), f(3), and f(−1).
Solvef(0) = 0 − 0 + 5 = 5 f(3) = 9 − 6 + 5 = 8 f(−1) = 1 + 2 + 5 = 8

Answer: f(0) = 5, f(3) = 8, f(−1) = 8

Example 1.4.2

Evaluating a function with an expression as input

ProblemLet g(x) = 3x − 4. Find g(x + 1).
ThinkReplace every occurrence of x with (x + 1) — treat the input as a placeholder.
Solveg(x + 1) = 3(x + 1) − 4 = 3x + 3 − 4 = 3x − 1

Answer: g(x + 1) = 3x − 1

Student Note

Substituting an expression as the input of a function is essential for composition (Section 1.11). Practice treating the input as a placeholder that can hold any expression, not just a number.

Section 1.5

Domain and Range

Two Algebraic Domain Restrictions

Denominators: any value making the denominator zero is excluded.

Even-index radicals: the expression under the radical must be ≥ 0.

If neither restriction applies, the domain is all real numbers: (−∞, ∞).

Example 1.5.1

Finding the domain of three functions

Problem(a) f(x) = 3x² − 5 (b) g(x) = 1/(x − 4) (c) h(x) = √(2x − 6)
Solve (a)No denominator, no radical. Domain: (−∞, ∞).
Solve (b)x − 4 = 0 when x = 4. Exclude it. Domain: (−∞, 4) ∪ (4, ∞).
Solve (c)2x − 6 ≥ 0 → x ≥ 3Domain: [3, ∞).

Answers: (a) (−∞, ∞) (b) (−∞, 4) ∪ (4, ∞) (c) [3, ∞)

Example 1.5.2

Domain and range from context

ProblemA truck travels at 50 mph. D(t) = 50t. The driver works at most 8 hours. State domain and range in context.
AnswerDomain: [0, 8] (time in hours). Range: [0, 400] (distance in miles).

The truck travels between 0 and 400 miles during a work period.

Section 1.6

Graphs as Function Representations

A function graph is more than a picture — it is a complete record of the function's behavior. From any graph you can read domain (horizontal extent), range (vertical extent), intercepts, zeros, intervals of increase and decrease, maximum/minimum values, and end behavior.

Vertical Line Test

A graph represents a function if and only if every vertical line crosses the graph at most once. If any vertical line crosses in two or more places, the graph is not a function — one input is producing multiple outputs.

Example 1.6.1

Reading domain, range, zeros, and behavior from a graph

SetupGraph: upward parabola with vertex (2, −3), x-intercepts near x ≈ 0.3 and x ≈ 3.7, y-intercept at (0, 1).
DomainGraph extends left and right without stopping → (−∞, ∞).
RangeLowest point is vertex at y = −3, graph rises from there → [−3, ∞).
Zerosx ≈ 0.3 and x ≈ 3.7 (where the graph meets the x-axis).
DecreasingGraph falls from left until vertex at x = 2. Decreasing on (−∞, 2).

Domain: (−∞, ∞) | Range: [−3, ∞) | Zeros: ≈ 0.3, ≈ 3.7 | Decreasing on (−∞, 2)

Section 1.7

Intercepts, Zeros, and Solutions

Three terms — intercept, zero, and solution — describe the same idea from different perspectives. Understanding their connection is essential throughout Algebra II.

Term	Definition	How to Find It
y-intercept	Where the graph crosses the y-axis	Evaluate f(0)
x-intercept	Where the graph crosses the x-axis	Solve f(x) = 0; record as point (x, 0)
Zero of f	A value x where f(x) = 0	x-coordinate of each x-intercept
Solution to f(x) = 0	Same as the zeros of f	Set f(x) = 0 and solve

Example 1.7.1

Finding intercepts and zeros

ProblemLet f(x) = x² − x − 6. Find the y-intercept and the zeros.
y-interceptf(0) = 0 − 0 − 6 = −6. The y-intercept is (0, −6).
Zerosx² − x − 6 = 0 (x − 3)(x + 2) = 0 x = 3 or x = −2
Checkf(3) = 9 − 3 − 6 = 0 ✓ | f(−2) = 4 + 2 − 6 = 0 ✓

y-intercept: (0, −6) | x-intercepts: (3, 0) and (−2, 0) | Zeros: x = 3, x = −2

Section 1.8

Average Rate of Change

For a linear function, the rate of change is constant. For every other function type, it varies. To describe how a function changes over a specific interval, use the average rate of change.

Average Rate of Change Formula

The average rate of change of f over [a, b] is:

(f(b) − f(a)) / (b − a)

This is the slope of the secant line connecting (a, f(a)) and (b, f(b)) on the graph of f.

Example 1.8.1

Computing average rate of change

ProblemLet f(x) = x² + 3. Find the average rate of change over [1, 4].
Solvef(1) = 1 + 3 = 4 f(4) = 16 + 3 = 19 (19 − 4) / (4 − 1) = 15 / 3 = 5
CheckOutput increases 15 units over a horizontal span of 3. A slope of 5 is reasonable for a parabola rising across this interval.

Answer: The average rate of change over [1, 4] is 5.

Example 1.8.2

Interpreting average rate of change in context

ProblemA store's monthly revenue R satisfies R(3) = 42 and R(9) = 66 (in thousands of dollars). Find and interpret the average rate of change from March to September.
Solve(66 − 42) / (9 − 3) = 24 / 6 = 4

Answer: Revenue increased by an average of $4,000 per month from March through September.

Bridge to Calculus

Average rate of change is the algebraic foundation for instantaneous rate of change, the central idea of Calculus I. What you are learning here — the slope of a secant line over an interval — leads directly to the derivative when the interval shrinks to a single point.

Section 1.9

Transformations of Functions

A transformation changes a function by shifting, stretching, compressing, or reflecting its graph. Understanding transformations means you can analyze any function in a family by starting from the parent and applying changes systematically.

Transformation	Form	Effect on Graph
Vertical shift up	`f(x) + k, k > 0`	Moves every point k units up
Vertical shift down	`f(x) + k, k < 0`	Moves every point \|k\| units down
Horizontal shift right	`f(x − h), h > 0`	Moves every point h units right
Horizontal shift left	`f(x − h), h < 0`	Moves every point \|h\| units left
Vertical stretch	`a · f(x), \|a\| > 1`	Taller; outputs scaled by \|a\|
Vertical compression	`a · f(x), 0 < \|a\| < 1`	Shorter; outputs scaled by \|a\|
Reflection over x-axis	`−f(x)`	Flips graph upside down
Reflection over y-axis	`f(−x)`	Flips graph left-right

General Transformed Form

g(x) = a · f(b(x − h)) + k

h → horizontal shift | k → vertical shift | a → vertical scale / reflection | b → horizontal scale / reflection

Example 1.9.1

Identifying all transformations from the general form

ProblemDescribe the transformations producing g(x) = −2(x + 3)² − 1 from f(x) = x².
ThinkRewrite as g(x) = −2(x − (−3))² + (−1) to read parameters: a = −2, h = −3, k = −1.
Solveh = −3 → shift 3 units LEFT k = −1 → shift 1 unit DOWN a = −2 → vertical STRETCH by factor 2, then REFLECT over x-axis
GraphDiagram instruction: draw f(x) = x² with vertex at origin. Draw g(x) = −2(x+3)² − 1 with vertex at (−3, −1), opening downward, narrower. Label both vertices.

Answer: Shift 3 left, 1 down; stretch vertically by factor 2; reflect over x-axis.

Student Warning

Horizontal shifts are opposite to the sign inside. The expression f(x + 3) means h = −3, shifting the graph 3 units to the LEFT, not right. Think: the function reaches the same output at x + 3 that it used to reach at x, so the graph is displaced left.

Section 1.10

Parent Functions

A parent function is the simplest, untransformed version of a function family. Knowing the parent means knowing the basic shape, domain, range, and behavior for an entire family. Every other member of the family is a transformation of the parent.

Constant

f(x) = c

Graph: horizontal line
Domain: (−∞, ∞) | Range: {c}

Linear

f(x) = x

Graph: line, slope 1, through origin
Domain & Range: (−∞, ∞)

Absolute Value

f(x) = |x|

Graph: V-shape, vertex at origin
Domain: (−∞, ∞) | Range: [0, ∞)

Quadratic

f(x) = x²

Graph: parabola, opens up, vertex at origin
Domain: (−∞, ∞) | Range: [0, ∞)

Square Root

f(x) = √x

Graph: half-curve from origin
Domain: [0, ∞) | Range: [0, ∞)

Cube Root

f(x) = ∛x

Graph: S-curve through origin
Domain & Range: (−∞, ∞)

Cubic

f(x) = x³

Graph: steeper S-curve through origin
Domain & Range: (−∞, ∞)

Reciprocal

f(x) = 1/x

Graph: two curves, opposite quadrants
Domain & Range: all reals except 0

Exponential

f(x) = bˣ, b > 0, b ≠ 1

Graph: growth or decay curve
Domain: (−∞, ∞) | Range: (0, ∞)

Logarithmic

f(x) = log_b(x)

Graph: slow-rising curve
Domain: (0, ∞) | Range: (−∞, ∞)

MGU Library Connection

A full parent function reference with labeled graphs, domain/range tables, and transformation summaries is in Appendix B: Function Parent Graphs. Students preparing for Precalculus may also consult the MGU Precalculus chapter overview for extensions of these families.

Section 1.11

Composition of Functions

Two functions can be composed: one function is applied first, and its output becomes the input of the second. Written (f ∘ g)(x) = f(g(x)). Order matters — composition is generally not commutative.

Composition

(f ∘ g)(x) = f(g(x))

Read "f of g of x." Compute g(x) first. Use that result as the input to f. Note: f(g(x)) and g(f(x)) are generally different functions.

Example 1.11.1

Computing both orders of composition

ProblemLet f(x) = 2x + 1 and g(x) = x² − 3. Find (f ∘ g)(x) and (g ∘ f)(x).
f ∘ gf(g(x)) = f(x² − 3) = 2(x² − 3) + 1 = 2x² − 5
g ∘ fg(f(x)) = g(2x + 1) = (2x + 1)² − 3 = 4x² + 4x − 2

(f ∘ g)(x) = 2x² − 5 | (g ∘ f)(x) = 4x² + 4x − 2 — Different functions, confirming non-commutativity.

Example 1.11.2

Evaluating a composition at a point

ProblemFind (f ∘ g)(4) using the functions above.
Method 1g(4) = 16 − 3 = 13 f(13) = 2(13) + 1 = 27
Method 2Use the formula: (f ∘ g)(4) = 2(4)² − 5 = 32 − 5 = 27 ✓

Answer: (f ∘ g)(4) = 27

Section 1.12

Inverse Relationships

If a function f takes input x and produces output y, the inverse function f⁻¹ takes y as input and returns x. The inverse reverses the operation of the original function.

Verifying Inverse Functions

Two functions f and g are inverses if and only if:

(f ∘ g)(x) = x AND (g ∘ f)(x) = x

Graphically: the graph of f⁻¹ is the reflection of the graph of f across the line y = x.

One-to-one test: if any horizontal line crosses the graph more than once, the function is not one-to-one and has no inverse function. This is the horizontal line test.

Example 1.12.1

Verifying inverse functions by composition

ProblemVerify that f(x) = 3x − 6 and g(x) = (x + 6)/3 are inverses.
f ∘ gf(g(x)) = f((x+6)/3) = 3·((x+6)/3) − 6 = (x + 6) − 6 = x ✓
g ∘ fg(f(x)) = g(3x − 6) = ((3x − 6) + 6)/3 = 3x/3 = x ✓

Both compositions equal x. The functions are inverses. ✓

Example 1.12.2

Finding an inverse algebraically

ProblemFind the inverse of f(x) = 2x + 5.
MethodReplace f(x) with y → swap x and y → solve for y → replace y with f⁻¹(x).
Solvey = 2x + 5 x = 2y + 5 x − 5 = 2y y = (x − 5)/2
Checkf(f⁻¹(x)) = 2·((x−5)/2) + 5 = (x − 5) + 5 = x ✓

Answer: f⁻¹(x) = (x − 5)/2

Student Warning

The notation f⁻¹(x) does NOT mean 1/f(x). The superscript −1 denotes the inverse function, not a reciprocal. This is one of the most common notational errors in Algebra II. Context will clarify which is meant, but never confuse them.

Section 1.13

Modeling with Functions

A mathematical model is a function or equation used to represent a real-world relationship. The first step is always recognizing which function family fits the pattern of the situation.

Pattern in the Situation	Best Model	General Form
Constant rate of change; additive growth or decay	Linear	`f(x) = mx + b`
Curved growth; projectile motion; area; optimization	Quadratic	`f(x) = ax² + bx + c`
Multiplicative growth or decay; constant percent change	Exponential	`f(x) = a · bˣ`
Inverse variation; one quantity decreases as other increases	Rational	`f(x) = k/x`

Example 1.13.1

Building a linear model

ProblemA taxi charges $2.50 per mile plus a $4.00 base fare. Write C(m). Find the cost for 7 miles.
SolveC(m) = 2.50m + 4 C(7) = 2.50(7) + 4 = 17.50 + 4 = 21.50

Answer: C(m) = 2.50m + 4. A 7-mile ride costs $21.50.

Example 1.13.2

Interpreting a quadratic model in context

ProblemA ball thrown from a 6-foot platform: h(t) = −16t² + 40t + 6. Find h(1) and h(2).
TypeQuadratic, degree 2, negative leading coefficient — downward parabola. Models projectile motion.
Solveh(1) = −16 + 40 + 6 = 30 feet h(2) = −64 + 80 + 6 = 22 feet

Ball starts at 6 ft, rises to 30 ft after 1 second, descends to 22 ft after 2 seconds.

MGU Library Connection

Mathematical modeling is used throughout MGU courses. Students in Physics, Finance, Engineering, and Computer Science will encounter function models regularly. The MGU Precalculus chapter on modeling extends these ideas. The MGU Statistics sequence introduces regression models for real data.

Section 1.14

Preparing for Advanced Function Families

Every concept in this chapter will reappear across every function family in Algebra II. The analytical tools are consistent — the families change.

Concept from Chapter 1	Where It Reappears in Algebra II
Transformations	Applied to radical, exponential, logarithmic, and trigonometric functions; Chapters 8–13
Domain and Range	Rational functions exclude division-by-zero values; log functions require positive input; Chapter 7 onward
Zeros and Intercepts	Polynomials may have many zeros; rational functions have both zeros and excluded values; Chapters 5–7
Composition	Multi-stage models; verifying inverse relationships; Precalculus chain rule foundation
Inverse Functions	Exponential and logarithmic functions are inverses of each other — central to Chapters 9–10
Modeling	Grows to include exponential, rational, and logarithmic models with real data; Chapter 14
Average Rate of Change	Compared across function families; foundation for the derivative in Calculus I

Chapter 1 — Back Matter

Chapter Summary

Sec 1.1Algebra II studies function families — polynomial, rational, radical, exponential, logarithmic, trigonometric — using consistent analytical tools: domain, range, transformations, zeros, inverses, and models.
Sec 1.2Expressions are built from terms with coefficients and variables. Evaluating means substituting. Simplifying means distributing and combining like terms. The same rules apply to complex expressions with radicals or rational exponents.
Sec 1.3Equations are solved by inverse operations. Inequalities require reversing the symbol when multiplying or dividing by a negative. Inequality solutions are intervals, not single values.
Sec 1.4f(x) means the output of function f when the input is x. Functions can be evaluated at numbers or expressions. Treating the input as a placeholder is essential for composition work later.
Sec 1.5Domain is restricted by zero denominators and even roots of negative numbers. Context may add further restrictions. Range is the set of all achievable outputs, often read from a graph.
Sec 1.6The vertical line test identifies function graphs. Graphs carry domain, range, intercepts, zeros, increase/decrease intervals, extrema, and end behavior — all readable without equations.
Sec 1.7Zeros, x-intercepts, and solutions to f(x) = 0 all refer to the same values. The y-intercept is found by evaluating f(0).
Sec 1.8Average rate of change over [a, b] equals (f(b) − f(a))/(b − a). It is the slope of the secant line and the algebraic foundation for instantaneous rate of change in Calculus.
Sec 1.9The general form g(x) = a·f(b(x − h)) + k encodes all four transformation types. Horizontal shifts move opposite to the sign of h inside the function.
Sec 1.10Ten parent functions serve as the base for every family: constant, linear, absolute value, quadratic, square root, cube root, cubic, reciprocal, exponential, and logarithmic.
Sec 1.11(f ∘ g)(x) = f(g(x)) applies g first, then f. Composition is not commutative. It is the structural basis of inverse verification and the calculus chain rule.
Sec 1.12f and g are inverses when both compositions return x. Finding an inverse: replace f(x) with y, swap x and y, solve for y. The graph of f⁻¹ is the reflection of f across y = x. Inverse functions require one-to-one originals.
Sec 1.13Modeling means choosing a function family that fits the pattern and using it to predict or interpret. Linear for constant rate; quadratic for curved motion; exponential for percent change; rational for inverse variation.
Sec 1.14Every concept from this chapter reappears across all fourteen chapters. The tools are consistent. The families change. Algebra II develops both the tools and the flexibility to apply them.

Key Terms Review

Average rate of change

The ratio (f(b) − f(a))/(b − a). Slope of the secant line over [a, b].

Composition

(f ∘ g)(x) = f(g(x)). Apply g first, then f. Generally not commutative.

Domain

The set of all allowable inputs for a function. Restricted by zero denominators and even roots of negatives.

End behavior

What a function's output does as x grows very large (→ +∞) or very small (→ −∞).

Excluded value

A value not in the domain — typically a zero of a denominator or a value making a radical negative.

Function

A rule assigning exactly one output to each input.

Function family

A group of related functions sharing the same general structure. Linear, quadratic, and exponential are examples.

Horizontal shift

Left or right movement from f(x − h). The direction is opposite to the sign of h.

Inverse function

f⁻¹ reverses f. Both compositions f(f⁻¹(x)) and f⁻¹(f(x)) equal x. Graph is reflected across y = x.

One-to-one function

Every output comes from exactly one input. Passes the horizontal line test. Required for an inverse to exist.

Parent function

The simplest untransformed member of a function family, with no shifts, stretches, or reflections.

Range

The set of all possible outputs of a function — the y-values actually produced.

Reflection

Flip of a graph across the x-axis (from −f(x)) or the y-axis (from f(−x)).

Secant line

The line connecting two points on a function graph. Its slope is the average rate of change.

Transformation

Any shift, stretch, compression, or reflection applied to a parent graph. Encoded in g(x) = a·f(b(x−h))+k.

Vertical line test

A graph is a function if no vertical line crosses it more than once.

Vertical shift

Up or down movement from f(x) + k. Positive k shifts up; negative k shifts down.

y-intercept

Where the graph meets the y-axis. Found by evaluating f(0).

Zero of a function

A value of x where f(x) = 0. Corresponds to an x-intercept and a solution to f(x) = 0.

Concept Review Questions

Answer in complete sentences. These questions check conceptual understanding, not just computation.

What does it mean to say that a function assigns "exactly one output to each input"? Give an example of a relationship that is not a function and explain why.
Why is the notation f(3) = 7 more precise than saying "when x = 3, the output is 7"?
Describe two types of algebraic restrictions that can limit the domain of a function. Give an example of each.
Explain the difference between a zero of a function and a y-intercept.
Why is average rate of change more useful than simply saying "the output increased"?
The transformation g(x) = f(x + 4) shifts the graph left by 4 units, not right. Explain why the shift goes left even though the sign inside is positive.
How do you determine whether a function has an inverse that is also a function?
What is the difference between f⁻¹(x) and 1/f(x)? Give a concrete example.
A student says (f ∘ g)(x) = (g ∘ f)(x) is always true. Is this correct? Explain with an example.
How does the function type influence which mathematical model to choose for a real situation?

Algebra I Review Practice

Set AExpressions and Equations

Evaluate 4x² − 3x + 1 when x = −2.
Simplify 5(2x − 3) − 2(4x + 1).
Solve: 4(x − 2) = 3x + 1.
Solve: x² − 7x + 10 = 0.
Solve and graph: −2x + 3 ≤ 9.
Factor: x² + 5x − 14.
Slope through (2, −1) and (−4, 5).
Line with slope −3 through (0, 4).
Is {(1,5),(2,8),(3,5),(4,11),(2,6)} a function? Explain.
Use the quadratic formula to solve 2x² − 3x − 2 = 0.

Set BFunction Notation and Composition

f(x) = x² − 4x + 3. Find f(0), f(5), f(−3), f(a), f(x+2).
g(x) = (2x−1)/(x+3). Find g(0), g(−1), g(3). State excluded values.
h(x) = √(x+5). State domain, then find h(4), h(0), h(−5).
C(0) = 150, C(50) = 450. Interpret each in context.
If f(2) = 7 and g(7) = 3, find (g ∘ f)(2).
f(x) = 3x + 2, g(x) = x². Find (f ∘ g)(x) and (g ∘ f)(x). Equal?

Set CTransformations

Describe transformations: g(x) = x² + 5 from f(x) = x².
Describe transformations: g(x) = (x − 3)² from f(x) = x².
Describe transformations: g(x) = −x² from f(x) = x².
Describe transformations: g(x) = 2(x+1)² − 4 from f(x) = x².
If f passes through (−2,1), (0,3), (4,−2), give corresponding points on y = f(x) + 6.
Same three points on y = f(x − 2).
Same three points on y = −f(x).
Write equation: parent f(x) = √x, shifted 4 right and 2 down.

Set DInverses and Modeling

Find f⁻¹(x) for f(x) = 4x − 8. Verify by composition.
Find f⁻¹(x) for f(x) = (x + 3)/5. Verify.
Car at 65 mph. Write D(t). Domain if trip ≤ 6 hours?
Phone plan: $35/month + $0.10/text. Write C(n). Find C(200) and interpret.
h(t) = −16t² + 100 (dropped object). Height at t = 1? When does it reach 36 ft? When does it hit the ground?
Population doubles every 3 hours starting from P₀. Write P(t).

Error Analysis

Each problem below contains a student error. Identify the mistake and write the correct solution.

Problem 1

A student evaluates f(x) = 3x² − 2 at x = −3.

f(−3) = 3(−3)² − 2 = 3(−9) − 2 = −29