Carey AP PrecalculusPractice Lab
Topic study & practice

2.8 · Inverse Functions

Understand the ideas before moving on. Explore distinctive knowledge notes, worked examples, and connections between representations.

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Topic 2.8 · Inverse Functions

Reverse input-output pairs on a one-to-one domain and verify the resulting inverse.

An inverse reverses input and output; it is not a reciprocal.

Teaching illustration · scroll to inspect
An inverse reverses input and output; it is not a reciprocal.

What a strong response shows

  • Check whether the original function is one-to-one.

  • Swap the input and output, then solve.

  • The original domain and range exchange roles.

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01

What You Need to Know

Key idea

An inverse reverses a function’s input-output pairs. A function needs a one-to-one domain for the inverse to be a function.

Fast decision rule

Swap xx and yy, solve for yy, and state any domain restriction needed for one-to-one behavior.

Worked steps

If a function sends 22 to 77, its inverse sends 77 to 22.

Common mistake

Writing a reciprocal instead of reversing the input-output rule.

Mini check

What happens to every ordered pair under an inverse?

02

Learning Objectives

Key idea

2.8.A — Determine the input-output pairs of the inverse of a function. 2.8.B — Determine the inverse of a function on an invertible domain.

Fast decision rule

Match the prompt verb to the representation it asks you to build or interpret.

Worked steps

11. Name the given quantities. 22. Choose the matching function or representation. 33. Show the parameter or operation that changes. 44. Check one value in context.

Common mistake

Starting calculations before identifying what the input and output mean.

Mini check

Can you state what the input, output, and requested evidence are before choosing a formula?

03

Key Concepts

Key idea

The graph of an inverse reflects across y=xy=x, and the domain and range switch roles.

Fast decision rule

Use the horizontal line test to decide whether the chosen domain is invertible.

Worked steps

Restrict a parabola to one side of its vertex, then solve the swapped equation on that domain.

Common mistake

Keeping the original domain after finding the inverse.

Mini check

Which test checks whether an inverse is a function?

04

Relevant Representations

Key idea

Pairs, tables, graphs, and equations show the same reversal from different angles.

Fast decision rule

Check one pair and one composition to confirm the inverse.

Inverse evidence
OriginalInverseCheck
(a, b)\left(a,\,b\right)(b, a)\left(b,\,a\right)coordinates swap
domainrangesets switch
ff graphreflected graphmirror over y=xy=x
f(f−1(x))f\left(f^{-1}\left(x\right)\right)xxcomposition check
Worked steps

Swap each table row, reflect the graph mentally across y=xy=x, and verify both compositions where defined.

Common mistake

Confusing an inverse function with the inverse relation when the original is not one-to-one.

Mini check

Which two sets switch when a function is inverted?

05

Essential Relationships or Formulas

Key idea

Use an exact relationship to expose the pattern before substituting values.

f−1(f(x))=xf^{-1}(f(x))=x
f(f−1(x))=xf(f^{-1}(x))=x
Fast decision rule

Keep the variables and restrictions visible, then substitute with units attached.

Worked example

Write the relationship, identify each parameter, substitute the known values, and check one output.

Common mistake

Using a familiar formula without checking its domain, base, interval, or unit assumptions.

Mini check

Which symbol or parameter carries the feature the prompt asks you to find?

06

Worked Examples

Key idea

A short, labeled chain of steps is easier to check than a page of unexplained algebra.

Fast decision rule

Write the representation first, calculate second, and interpret last.

Worked steps

Swap a pair, solve the swapped equation, and verify with a composition or reflection check.

Common mistake

Skipping the interpretation step and reporting an answer with no units, domain, or model evidence.

Mini check

Which step would you show so another student could reproduce the result?

07

Common Mistakes

Key idea

Most errors come from a mismatch between the pattern, the operation, and the allowed input.

Fast decision rule

Before submitting, check pattern, operation, domain, and units in that order.

Worked steps

Circle the first line where the pattern or restriction changes. Correct that line, then recompute only the affected steps.

Common mistake

Trusting a calculator output that hides a wrong model, sign, base, or domain.

Mini check

What independent check would catch your most likely error?

08

AP Exam Tips

Key idea

Use the topic workflow to connect the representation, the equation, and the context before calculating.

Fast decision rule

Read the prompt, choose the smallest reliable representation, and verify the result in context.

Worked steps

11. Identify the pattern. 22. Write the matching form. 33. Show the key substitution or transformation. 44. Check units, domain, and direction.

Common mistake

Submitting a number without explaining why the selected model or transformation fits.

Mini check

Can you name the evidence that supports your final model or answer?

Practice

Open this topic’s practice set to apply the method. Work independently, then compare the evidence in your reasoning with the checked explanation.

TRY IT IN A NEW SETTING

Can you transfer the method?

Find the inverse of f(x)=2x+3f(x)=2x+3 and verify one composition.

Compare your reasoning

Swap x=2y+3x=2 y+3 and solve y=x−32y=\frac{x-3}{2}. Then f(x−32)=xf\left(\frac{x-3}{2}\right)=x for every real xx. The inverse is not the reciprocal 12x+3\frac{1}{2 x+3}.

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

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