inverse of linear functions common core algebra 2 homework

Common Core Algebra II.Unit 2.Lesson 6.Inverse Functions

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Sep 22, 2016

Learning Common Core Algebra II Unit 2 Lesson 6 Inverse Functions by EmathInstruction

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Inverse Functions Worksheet

Students will practice work with inverse functions including identifying the inverse functions , graphing inverses and more.

Example Questions

Directions: Find the inverse of each function

Challenge Problems. Part II

• Inverse Function
• Relations and Functions -- everything you might want to know.
• Domain and Range
• Functions and Relations in Math

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Inverse of Linear Function

Finding the inverse of a linear function.

The inverse of a linear function is much easier to find as compared to other kinds of functions such as quadratic and rational. The reason is that the domain and range of a linear function naturally span all real numbers unless the domain is restricted.

Before I go over five (5) examples to illustrate the procedure, I want to show you how the domain and range of a given function and its inverse are related.

Domain and Range are just swapped!

Notes to consider about the diagram :

• The domain of the original function becomes the range of the inverse function.
• The range of the original function becomes the domain of the inverse function.
• It is customary to use the letter [latex]\large{\color{blue}x}[/latex] for the  domain  and [latex]\large{\color{red}y}[/latex] for the  range .

The general approach on how to algebraically solve for the inverse is as follows:

Key Steps in Finding the Inverse of a Linear Function

• Replace [latex]f\left( x \right)[/latex] by [latex]y[/latex].
• Switch the roles of [latex]x[/latex] and [latex]y[/latex], in other words, interchange [latex]x[/latex] and [latex]y[/latex] in the equation.
• Solve for [latex]y[/latex] in terms of [latex]x[/latex].
• Replace [latex]y[/latex] by [latex]{f^{ – 1}}\left( x \right)[/latex] to get the inverse function.
• Sometimes, it is helpful to use the domain and range of the original function to identify the correct inverse function out of two possibilities. This happens when you get a “plus or minus” case in the end.

Examples of How to Find the Inverse of a Linear Function

Example 1: Find the inverse of the linear function

This function behaves well because the domain and range are both real numbers. This ensures that its inverse must be a function too. Maybe you’re familiar with the Horizontal Line Test which guarantees that it will have an inverse whenever no horizontal line intersects or crosses the graph more than once.

Use the key steps above as a guide to solve for the inverse function:

That was easy!

Example 2: Find the inverse of the linear function

Towards the end part of the solution, I want to make the denominator positive so it looks “good”. I did it by multiplying both the numerator and denominator by [latex]-1[/latex].

Example 3: Find the inverse of the linear function

Some students may consider this as a rational function because the equation contains some rational expressions. Don’t be confused by the fractions here. Yes, it has fractions however there are no variables in the denominator. This makes it just a regular linear function.

To work this out, I must get rid of the denominator. I will accomplish that by multiplying both sides of the equation by their Least Common Denominator (LCD).

As shown above, you can write the final answers in two ways. One with a single denominator, and the other is decomposed into partial fractions.

Example 4: Find the inverse of the linear function below and state its domain and range.

This is a “normal” linear function, however, with a restricted domain. The allowable values of [latex]x[/latex] start at [latex]x=2[/latex] and go up to positive infinity. The range can be determined using its graph. Remember that range is the set of all y values when the acceptable values of x (domain) are substituted into the function.

Pay particular attention to how the domain and range are determined using its graph.

Finding the inverse of this function is really easy. But keep in mind how to correctly describe the domain and range of the inverse function. We have gone over this concept at the beginning of this section about the swapping of domain and range .

Always verify the domain and range of the inverse function using the domain and range of the original. They are just interchanged.

Example 5: Find the inverse of the linear function below and state its domain and range.

The first step is to plot the function in [latex]xy[/latex]-axis. Clearly label the domain and the range.

Open circle (unshaded dot) means that the number at that point is excluded. If you need to refresh on this topic, check my separate lesson about Solving Linear Inequalities .

Secondly, find the inverse algebraically using the suggested steps. Make sure that you write the correct domain and range of the inverse function.

The [latex]x[/latex] variable in the original equation has a coefficient of [latex]-1[/latex]. Keep track of this as you solve for the inverse.

I hope that you gain some basic ideas on how to find the inverse of a  linear function . I recommend that you survey the related lessons on how to find inverses of other types of functions.

You might also like these tutorials:

• Inverse of a 2×2 Matrix
• Inverse of Absolute Value Function
• Inverse of Constant Function
• Inverse of Exponential Function
• Inverse of Logarithmic Function
• Inverse of Quadratic Function
• Inverse of Rational Function
• Inverse of Square Root Function

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Unit 2 – Linear Expressions, Equations, and Inequalities

This unit is all about linear topics, which is a major focus of Common Core Algebra I.  We develop general methods for solving linear equations using properties of equality and inverse operations.  Thorough review is given to review of equation solving from Common Core 8th Grade Math. Solutions to equations and inequalities are defined in terms of making statements true.  This theme is emphasized throughout the unit.  Modeling with both linear equations and inequalities is stressed.

Equations and Their Solutions

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

Seeing Structure to Solve Equations

A Linear Equation Solving Review

Justifying Steps in Solving an Equation

Linear Word Problems

More Linear Equations and Consecutive Integer Games

Solving Linear Equations with Unspecified Constants

Inequalities

Solving Linear Inequalities

Compound Inequalities

More Work with Compound Inequalities

Interval Notation

Modeling with Inequalities

Unit Review

Unit #2 Review – Linear Equations and Inequalities

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit #2 Assessment Form A

EDITABLE ASSESSMENT

Unit #2 Assessment Form B

Unit #2 Assessment Form C

Unit #2 Assessment Form D

Unit #2 Exit Tickets

Unit #2 Mid-Unit Quiz (Through Lesson #6) – Form A

Unit #2 Mid-Unit Quiz (Through Lesson #6) – Form B

Unit #2 Mid-Unit Quiz (Through Lesson #6) – Form C

U02.AO.01 – Basic Equation Solving

EDITABLE RESOURCE

U02.AO.02 – More Work with Linear Word Problems (After Lesson #7)

U02.AO.03 – Inequality Warm-Up Set (Prior to Lesson #8)

U02.AO.04 – Equation and Inequality Practice

U02.AO.05 – Using Structure to Understand Solutions to Equations (Enrichment)

U02.AO.06 – Solving Linear Equations – Extra Practice

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