Learn on PengiCalifornia Reveal Math, Algebra 1Unit 2: Relations and Functions

2-2 Functions

In this Grade 9 lesson from California Reveal Math Algebra 1 (Unit 2: Relations and Functions), students learn to identify whether a relation is a function by applying the definition that each element of the domain must be paired with exactly one element of the range. Students practice using mapping diagrams, ordered pairs, tables, and the vertical line test to classify relations as functions or non-functions. The lesson also introduces function notation f(x) and guides students through evaluating and interpreting function values in real-world contexts.

Section 1

Repeated Outputs vs. Repeated Inputs: What Breaks a Function?

Property

A relation is not a function only when the same input maps to two or more different outputs.

Repeated outputs are perfectly allowed:

Section 2

The Vertical Line Test

Property

The vertical line test determines if a graph represents a function by checking whether any perfectly vertical line intersects the graph at more than one point. If every vertical line intersects the graph at most once, then the graph represents a function.

Examples

An upward-opening parabola passes the vertical line test because each vertical line intersects it at most once, meaning it is a function.
A horizontal straight line passes the vertical line test because each vertical line intersects it exactly once.
A circle fails the vertical line test because a vertical line drawn through its center will intersect the circle at two different points (a top point and a bottom point).

Explanation

The vertical line test works because functions must have exactly one output (y-value) for each input (x-value). When a vertical line hits a graph at multiple points, it proves that a single x-value is producing multiple y-values, breaking the ultimate rule of a function. This visual trick gives you a split-second answer!

Section 3

Domain and Range from a Graph

Property

For a graph of a relation, the domain is the complete set of $x$ -values the graph covers (its horizontal extent), and the range is the complete set of $y$ -values the graph covers (its vertical extent).

\text{Domain} = \{x \mid x \text{ is covered by the graph left-to-right}\}

\text{Range} = \{y \mid y \text{ is covered by the graph bottom-to-top}\}

Section 4

Function Notation

Property

We use a letter like $f$ or $g$ to name a function. The notation $f(x)$ , read ' $f$ of $x$ ', represents the output value of the function $f$ when the input is $x$ . If $y$ is the output variable, we can write $y = f(x)$ . The parentheses in $f(x)$ do not indicate multiplication.

Function Notation:
Input variable
↓
$f(x) = y$
↑
Output variable

Examples

Instead of writing 'the area $A$ for a radius $r$ is $\pi r^2$ ', we can write $A(r) = \pi r^2$ . The notation $A(3)$ asks for the area of a circle with a radius of 3.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Repeated Outputs vs. Repeated Inputs: What Breaks a Function?

Property

A relation is not a function only when the same input maps to two or more different outputs.

Repeated outputs are perfectly allowed:

Section 2

The Vertical Line Test

Property

Examples

An upward-opening parabola passes the vertical line test because each vertical line intersects it at most once, meaning it is a function.
A horizontal straight line passes the vertical line test because each vertical line intersects it exactly once.
A circle fails the vertical line test because a vertical line drawn through its center will intersect the circle at two different points (a top point and a bottom point).

Explanation

Section 3

Domain and Range from a Graph

Property

\text{Domain} = \{x \mid x \text{ is covered by the graph left-to-right}\}

\text{Range} = \{y \mid y \text{ is covered by the graph bottom-to-top}\}

Section 4

Function Notation

Property

Function Notation:
Input variable
↓
$f(x) = y$
↑
Output variable

Examples

Instead of writing 'the area $A$ for a radius $r$ is $\pi r^2$ ', we can write $A(r) = \pi r^2$ . The notation $A(3)$ asks for the area of a circle with a radius of 3.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Repeated Outputs vs. Repeated Inputs: What Breaks a Function?

Property

A relation is not a function only when the same input maps to two or more different outputs.

Repeated outputs are perfectly allowed:

Section 2

The Vertical Line Test

Property

Examples

An upward-opening parabola passes the vertical line test because each vertical line intersects it at most once, meaning it is a function.
A horizontal straight line passes the vertical line test because each vertical line intersects it exactly once.
A circle fails the vertical line test because a vertical line drawn through its center will intersect the circle at two different points (a top point and a bottom point).

Explanation

Section 3

Domain and Range from a Graph

Property

\text{Domain} = \{x \mid x \text{ is covered by the graph left-to-right}\}

\text{Range} = \{y \mid y \text{ is covered by the graph bottom-to-top}\}

Section 4

Function Notation

Property

Function Notation:
Input variable
↓
$f(x) = y$
↑
Output variable

Examples

Instead of writing 'the area $A$ for a radius $r$ is $\pi r^2$ ', we can write $A(r) = \pi r^2$ . The notation $A(3)$ asks for the area of a circle with a radius of 3.