Lesson 14: Introduction to Functions
A function is a special type of relation where each input has exactly one output. In other words, a function takes an input value (usually called x) and assigns it to exactly one output value (usually called y).
Functions are often written as \( f(x) \), which is read as "f of x." The variable inside the parentheses is the input, and the expression after the equals sign is the rule.
Examples of Functions
- \( f(x) = 2x + 3 \): Input a number, double it, then add 3.
- \( g(x) = x^2 \): Input a number, then square it.
- \( h(x) = x - 5 \): Input a number, subtract 5.
Ways to Represent Functions
Functions can be represented in multiple ways:
- Words: "Take a number, multiply it by 2, then add 3."
- Equation: \( f(x) = 2x + 3 \).
- Table:
| x (input) | f(x) (output) |
|---|---|
| 0 | 3 |
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
What is Not a Function?
A relation is not a function if one input is paired with more than one output. For example:
- \( x = 2 \) gives both \( y = 3 \) and \( y = -3 \).
- This would fail the vertical line test on a graph (a vertical line crosses the graph in more than one place).
Practice Problems
- Is the relation \( \{(1,2), (2,4), (3,6), (4,8)\} \) a function? Why or why not?
- Evaluate \( f(x) = 3x - 2 \) when \( x = 5 \).
- Write a function in words and then as an equation.
Lesson Summary
- A function pairs each input with exactly one output.
- Functions can be written in words, equations, tables, or graphs.
- If an input has more than one output, it is not a function.