Learn on PengienVision, Algebra 1Chapter 10: Working With Functions

Lesson 7: Inverse Functions

In this Grade 11 enVision Algebra 1 lesson from Chapter 10, students learn how to find and use inverse functions, including the notation f⁻¹, by switching input and output values, solving algebraically, and graphing reflections across the line y = x. The lesson covers key conditions like one-to-one functions and restricted domains, and applies inverse functions to real-world problem-solving contexts.

Section 1

One-to-One Functions

Property

A function is one-to-one if each value in the range corresponds to one element in the domain.
For each ordered pair in the function, each $y$ -value is matched with only one $x$ -value. There are no repeated $y$ -values.

Horizontal Line Test
If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.

Examples

The set {(1, 2), (3, 4), (5, 6)} is a one-to-one function because each $y$ -value (2, 4, 6) is paired with only one $x$ -value.

Section 2

Definition of Inverse Functions

Property

Two functions are inverse functions if each one undoes the effect of the other.
The graphs of inverse functions are symmetric about the line $y = x$ .
If we interchange the variables in the function, we get an equivalent formula for its inverse.
For example, $y = \sqrt[3]{x}$ if and only if $x = y^3$ .

Examples

The inverse of taking the fifth power of a number is taking the fifth root. If we start with $x=2$ , taking the fifth power gives $2^5=32$ . The fifth root of 32 is $\sqrt[5]{32}=2$ , our original number.

The functions $f(x) = x+7$ and $g(x) = x-7$ are inverses. If you take a number, say 20, then $f(20) = 27$ . Applying the inverse gives $g(27) = 20$ , returning to the start.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

One-to-One Functions

Property

Horizontal Line Test
If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.

Examples

The set {(1, 2), (3, 4), (5, 6)} is a one-to-one function because each $y$ -value (2, 4, 6) is paired with only one $x$ -value.

Section 2

Definition of Inverse Functions

Property

Examples

The inverse of taking the fifth power of a number is taking the fifth root. If we start with $x=2$ , taking the fifth power gives $2^5=32$ . The fifth root of 32 is $\sqrt[5]{32}=2$ , our original number.

The functions $f(x) = x+7$ and $g(x) = x-7$ are inverses. If you take a number, say 20, then $f(20) = 27$ . Applying the inverse gives $g(27) = 20$ , returning to the start.

Overview

Lesson overview

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

Overview

Section 1

One-to-One Functions

Property

Horizontal Line Test
If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.

Examples

The set {(1, 2), (3, 4), (5, 6)} is a one-to-one function because each $y$ -value (2, 4, 6) is paired with only one $x$ -value.

Section 2

Definition of Inverse Functions

Property

Examples

The inverse of taking the fifth power of a number is taking the fifth root. If we start with $x=2$ , taking the fifth power gives $2^5=32$ . The fifth root of 32 is $\sqrt[5]{32}=2$ , our original number.

The functions $f(x) = x+7$ and $g(x) = x-7$ are inverses. If you take a number, say 20, then $f(20) = 27$ . Applying the inverse gives $g(27) = 20$ , returning to the start.