Lesson 6: One-to-One Functions and Their Inverses

What does it mean for a function to be one-to-one?

How can we determine if a function is one-to-one using a graph?

What is an inverse function, and how is it related to the original function?

How can we find the inverse of a function algebraically?

How are the graphs of a function and its inverse related geometrically?

Why do some functions need domain restrictions in order to have inverses?

One-to-one function

Horizontal Line Test

Inverse function

Inverse notation

Domain

Range

Reflection

Composition of functions

Identity function

Domain restriction

HSF-BF.B.4

Find inverse functions.

Verify inverses using composition.

HSF-IF.C.7

Graph functions and show key features.

HSF-BF.B.3

Identify the effect on the graph of replacing a function with its inverse.

Lesson Description

Students explore the concept of one-to-one functions and learn that only these functions possess inverses.

Students will:

Determine whether a function is one-to-one using the Horizontal Line Test

Find inverse functions algebraically

Verify inverse relationships using function composition

Graph functions and their inverses to observe symmetry across the line y=x

Students will analyze how domain and range switch roles between a function and its inverse.

Purpose

The purpose of this lesson is to help students understand the reversible nature of certain functions.

This concept is foundational for later topics including:

Exponential and logarithmic functions

Trigonometric inverses

Solving equations involving functions

Higher-level mathematics including calculus

Students also strengthen their understanding of function notation, graph interpretation, and algebraic manipulation.

Primary Level: DOK 2 – Skills and Concepts

Support for Developing Learners

Provide step-by-step guided examples for finding inverses.

Use visual graphs showing reflections across the line y=x.

Review solving equations and isolating variables.

Provide partially completed algebraic steps.

On-Level Strategies

Students determine whether functions are one-to-one using graphs.

Practice algebraic steps to find inverse functions.

Graph original and inverse functions using technology.

Enrichment / Extension

Explore domain restrictions required to make functions invertible.

Investigate inverses of quadratic or cubic functions with restrictions.

Connect inverse functions to real-world scenarios involving reversing processes.

Exit slips and quiz

textbook

• Desmos Graphing Calculator

• TI-83 / TI-84 Graphing Calculators

• SmartBoard graph demonstrations showing reflections across y=xy=x