Finding Inverse Functions Algebraically

Lesson 1 of 1: Swap-and-Solve Method

In this lesson:

• Use the swap-and-solve method to find inverse functions
• Find inverses of linear, power, and rational functions
• Verify the inverse using

Learning Objectives for This Lesson

You will:

1. Solve by isolating
2. Use swap-and-solve to find inverses
3. Find inverses of linear functions
4. Find inverses of power functions; restrict domain if needed
5. Find inverses of rational functions via cross-multiplication
6. Verify:

Solving and Finding

For :

• Solve :
• Using the inverse: , so

Both approaches give the same answer — they're the same operation.

The Swap-and-Solve Method: Four Steps

To find algebraically:

1. Replace with
2. Swap and
3. Solve for
4. Rename: write

Worked Example:

Step 1: Replace:

Step 2: Swap:

Step 3: Solve for :

Step 4: Rename:

Verifying the Inverse Function Property

Always check the inverse by composition:

If the composition doesn't simplify to , there's an algebra error.

Guided Example: Finding a Linear Inverse

Step 1: Replace with :

Step 2: Swap:

Step 3: Solve for — you complete this step

Step 4: Write

Then verify: substitute into and check you get .

Quick Check: Meaning of the Swap Step

The swap step () represents:

A. An algebraic trick to make the equation easier
B. Reversing the roles of input and output
C. Moving terms from one side of the equation to the other
D. Changing the domain of the function

Which answer best captures the mathematical meaning?

Practice Problems: Linear Inverse Functions

Find for each, then verify:

Use all four steps. Verify each by computing .

Answers to Linear Inverse Practice

1. :
2. :
3. :

Verify #3:

Power Function Inverses: Undoing

For , the inverse is

• Cubing and cube-rooting are natural inverses — no domain restriction needed
• Squaring and square-rooting require domain restriction to

Odd roots work on all reals; even roots need a positive input.

Finding the Inverse of a Cubic Function

Step 1:

Step 2: Swap:

Step 3: Solve:

Step 4:

Verify:

Domain Restriction for

is not one-to-one on all reals: and

With restriction :

Without restriction, two inputs give the same output — the inverse is not a function.

Worked: ,

Step 1: ,

Step 2: Swap: ,

Step 3: Solve: (positive root only, since )

Step 4:

Domain of : (range of original must be non-negative)

Quick Check: Power Function Inverse

• Find
• Then use to solve

Two steps: find the inverse, then evaluate it.

Practice Problems: Power Function Inverses

Find for each:

3. ,

For #3: state the domain of both and .

Answers to Power Function Practice

1. :
2. :
3. , : , domain ; also from restriction → domain:

Rational Function Inverses: Strategy First

For , the swap step leaves a fraction:

Strategy: cross-multiply, then collect all terms on one side, then factor and divide.

Finding the Inverse of a Rational Function

Step 2 (swap):

Step 3 (solve): Cross-multiply, collect terms, factor, divide:

Surprising Result: Is Its Own Inverse

This function is self-inverse (an involution): applying it twice returns the original input.

Verify:

Quick Check: Verify the Self-Inverse

For , verify that .

Work through the compound fraction step by step — don't skip any algebra.

Solving Using the Inverse

Solving is the same as computing .

Example: . Solve .

Find the inverse once; use it for any .

Practice Problems: Mixed Inverse Functions

For each, find , then use it to solve :

1. ; solve
2. ; solve
3. ; solve
4. ; solve

Answers to Mixed Inverse Practice

1. ;
2. ;
3. ;
4. ;

Key Takeaways from This Lesson

Swap-and-solve: Replace with , swap and , solve for , rename.

Always verify: — algebra errors show up here.

Domain restriction: Even-power functions need before inverting.

Watch out: Final answer must be in terms of only. Check one-to-one before starting.

Up Next: More Inverse Function Skills

Coming up in this cluster:

• Verify inverses by composition (HSF.BF.B.4.b)
• Read inverse values from graphs and tables (HSF.BF.B.4.c)
• Restrict domain to produce an invertible function (HSF.BF.B.4.d)
• Exponential-log as an inverse pair (HSF.BF.B.5)

You can now find inverse functions algebraically.