BetterGrades Precalculus · Unit 4 · Lesson

Graphs and verification of inverses

Verify inverse functions using composition and graph symmetry across y=x.

Opening

Start with the situation

Inverse functions undo each other in both composition orders and reflect across $y=x$ .

Many real models are built in stages—a conversion followed by a cost rule, or a measurement followed by a calibration. Composition records that order, while inverse reasoning asks whether the stages can be undone.

Before you begin

Prerequisite check

Evaluate function notation.
Determine domains from formulas.
Solve equations for a selected variable.

Core explanation

Explanation

Compute $f(f^{-1}(x))$ and $f^{-1}(f(x)),$ preserving their valid domains, and compare graph point pairs.

The two compositions begin on different domains: the original range and original domain.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through inverse reflection, composition loop, or another equivalent representation.

Conceptual reading

What the idea is really doing

Composition and inversion are about information flow. A composite sends an input through stages in a fixed order; an inverse reverses that flow only when each output identifies a unique input.

This lesson narrows that lens to one goal: verify inverse functions using composition and graph symmetry across $y=x$ . The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Compute $f(f^{-1}(x))$ .
$F^{-1}(f(x))$ .
Preserving their valid domains.
Compare graph point pairs.

Verification: Name the intermediate quantity, enforce its domain, and verify an inverse with composition. Units are especially useful because the output unit of one stage must match the input unit of the next.

Foundation walkthrough

Plan before calculating

Problem

Verify $f=3x-2 \qquad g=\frac{x+2}{3}$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Compute $f(f^{-1}(x))$ and $f^{-1}(f(x)),$ preserving their valid domains, and compare graph point pairs.
Conclusion: Both compositions simplify to $x$ .
Why the check works: The functions undo each other.

Worked examples

See the idea in three forms

foundation example

Verify $f=3x-2 \qquad g=\frac{x+2}{3}$

SolutionBoth compositions simplify to $x$ .

The functions undo each other.

representation example

Reflect $(3,-1)$ across $y=x$ .

Solution $(-1,3)$

This example expresses graphs and verification of inverses in a second form.

transfer example

Inverse range if original domain $[2,∞)$ .

Solution $[2,∞)$

The two compositions begin on different domains: the original range and original domain.

Inverse reflection. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The functions undo each other. — **Anchor figure · Inverse reflection**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The functions undo each other.

Composition loop. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for graphs and verification of inverses. — **Mechanism figure · Composition loop**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for graphs and verification of inverses.

Domain-range swap. Compare the valid path with the tempting shortcut. The figure shows why simplifying sqrt(x^2) to x without the branch restriction leads to a false conclusion. — **Comparison and error figure · Domain-range swap**
Compare the valid path with the tempting shortcut. The figure shows why simplifying $sqrt(x^2)$ to $x$ without the branch restriction leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is simplifying $sqrt(x^2)$ to $x$ without the branch restriction.

Check yourself

Verify $x+5 \qquad x-5$

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Practice

Ten concrete questions

Practice 101

Verify $x+5 \qquad x-5$

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Practice 202

Reflect $(3,-1)$ across $y=x$ .

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Practice 303

Inverse range if original domain $[2,∞)$ .

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Practice 404

Reflection line.

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Practice 505

Explain why this conclusion is valid: Both compositions simplify to $x$ . Use the foundation problem as evidence: Verify $f=3x-2$ and $g=\frac{x+2}{3}$ .

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Practice 606

Solve the representation example, then name the feature of graphs and verification of inverses that it illustrates: Reflect $(3,-1)$ across $y=x$ .

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is simplifying $sqrt(x^2)$ to $x$ without the branch restriction.

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Practice 808

Connect two representations for this example: Verify $f=3x-2$ and $g=\frac{x+2}{3}$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Inverse range if original domain $[2,∞)$ . Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for graphs and verification of inverses, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Domain restrictions and radical inverses, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Utah College Algebra
AP Precalculus framework

No long source passage is reproduced.

Verifyf=3x−2g=x+23f=3x-2 \qquad g=\frac{x+2}{3}f=3x−2g=3x+2​

Verifyf=3x−2g=x+23f=3x-2 \qquad g=\frac{x+2}{3}f=3x−2g=3x+2​

Reflect (3,−1)(3,-1)(3,−1) across y=xy=xy=x.

Inverse range if original domain [2,∞)[2,∞)[2,∞).

Original BetterGrades manuscript, rights-separated references.

Verify $f=3x-2 \qquad g=\frac{x+2}{3}$

Verify $f=3x-2 \qquad g=\frac{x+2}{3}$

Reflect $(3,-1)$ across $y=x$ .

Inverse range if original domain $[2,∞)$ .