Functions and Relations

Grade 9 · Math · Study Guide

Understand what makes a relation a function, use function notation, and evaluate functions for given inputs.

A relation is any set of ordered pairs. A function is a special relation where each input (x-value) maps to exactly one output (y-value). Think of a function as a machine: you put a number in, you get exactly one number out. The vertical line test is a quick way to check if a graph represents a function — if any vertical line crosses the graph more than once, it is not a function.

Function notation uses f(x) instead of y. For example, f(x) = 3x − 2 means the rule is 'multiply by 3 and subtract 2.' To evaluate f(4), substitute 4 for x: f(4) = 3(4) − 2 = 10. You can also find f(−1) = 3(−1) − 2 = −5.

The domain is the set of all valid input values (x), and the range is the set of all output values (f(x) or y). For most polynomial functions, the domain is all real numbers. For real-world functions, the domain may be restricted — you can't have a negative number of students or a negative time.

Function vs. Not a Function
Input (x)	Output (y)	Is it a function?
1 → 5	Each x has one y	Yes
2 → 7, 2 → 9	x = 2 maps to two different y-values	No
3 → 4, 5 → 4	Two x-values share one y	Yes (still OK)

Key terms

Relation: Any set of ordered pairs linking inputs to outputs
Function: A relation where every input has exactly one output
Function notation: f(x) read 'f of x'; replaces y and shows which variable is the input
Domain: The set of all valid input values (x-values)
Range: The set of all possible output values (y-values or f(x) values)
Vertical line test: If any vertical line hits the graph more than once, the graph is not a function

Flashcards

Function: Every input (x) maps to exactly one output (y)
Function notation f(x): Read 'f of x'; the output when x is the input
Domain: All valid input (x) values for a function
Range: All possible output (y) values of a function
Vertical line test: A graph is a function if no vertical line crosses it more than once
Evaluating a function: Substituting a specific value for x to find the output

Practice questions

If f(x) = 2x + 3, what is f(5)?
1. 10
2. 13
3. 8
4. 16
Show answer
Answer: 13
f(5) = 2(5) + 3 = 10 + 3 = 13.
A relation where x = 3 maps to both y = 1 and y = 7 is a function.
Show answer
Answer: False
One input (x = 3) mapping to two outputs violates the definition of a function.
The set of all input values of a function is called the ___.
Show answer
Answer: domain
The domain is all allowable inputs; the range is all resulting outputs.
Which set of ordered pairs represents a function?
1. {(1,2),(1,3),(2,4)}
2. {(1,2),(2,3),(3,4)}
3. {(2,5),(2,6),(3,7)}
4. {(4,1),(4,2),(4,3)}
Show answer
Answer: {(1,2),(2,3),(3,4)}
Each x-value appears only once, so each input has exactly one output.
If g(x) = x² − 1, what is g(−3)?
1. 8
2. 10
3. −10
4. −8
Show answer
Answer: 8
g(−3) = (−3)² − 1 = 9 − 1 = 8.

Fun facts

Functions are everywhere: your phone's battery percentage is a function of time, and a recipe is a function — input ingredients, output a dish.
The notation f(x) was introduced by Swiss mathematician Leonhard Euler in the 18th century.

Get a personalized course on Functions and Relations built around your child.