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BetterGrades Precalculus · Unit 4 · Lesson

Constructing inverse functions

Construct an inverse function algebraically and state the swapped domain and range.

Opening

Start with the situation

An inverse function reverses input and output roles; it is not the reciprocal 1f\frac{1}{f}.

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

Write y=f(x),y=f(x), exchange xx and y, solve for y, and state the swapped domain and range.

The original function must be one-to-one on its stated domain, and a restricted branch determines any root sign.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through coordinate reversal, algebraic inversion, 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: construct an inverse function algebraically and state the swapped domain and range. 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

  1. Write y=f(x)y=f(x).
  2. Exchange xx.
  3. Y.
  4. Solve for yy.

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

Find inverse of3x−73x-7

Plan
Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Write y=f(x),y=f(x), exchange xx and y, solve for y, and state the swapped domain and range.
Conclusion
f−1(x)=x+73f^{-1}(x)=\frac{x+7}{3}
Why the check works
The inverse reverses the operations in reverse order.
Worked examples

See the idea in three forms

foundation example

Find inverse of3x−73x-7

Solutionf−1(x)=x+73f^{-1}(x)=\frac{x+7}{3}

The inverse reverses the operations in reverse order.

representation example

Inverse of 4x4x.

Solutionx4\frac{x}{4}

This example expresses constructing inverse functions in a second form.

transfer example

Inverse of x+32\frac{x+3}{2}.

Solution2x−32x-3

The original function must be one-to-one on its stated domain, and a restricted branch determines any root sign.

Coordinate reversal. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The inverse reverses the operations in reverse order.
Read this graph as text

Constructing inverse functions · Coordinate reversal. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The inverse reverses the operations in reverse order. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Construct an inverse function algebraically and state the swapped domain and range.

Anchor figure · Coordinate reversal

Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The inverse reverses the operations in reverse order.

Algebraic inversion. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for constructing inverse functions.
Read this graph as text

Constructing inverse functions · Algebraic inversion. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for constructing inverse functions. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Construct an inverse function algebraically and state the swapped domain and range.

Mechanism figure · Algebraic inversion

Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for constructing inverse functions.

Inverse versus reciprocal. Compare the valid path with the tempting shortcut. The figure shows why keeping both plus and minus branches and producing an inverse relation rather than a function leads to a false conclusion.
Read this graph as text

Constructing inverse functions · Inverse versus reciprocal. Compare the valid path with the tempting shortcut. The figure shows why keeping both plus and minus branches and producing an inverse relation rather than a function leads to a false conclusion. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Construct an inverse function algebraically and state the swapped domain and range.

Comparison and error figure · Inverse versus reciprocal

Compare the valid path with the tempting shortcut. The figure shows why keeping both plus and minus branches and producing an inverse relation rather than a function leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is keeping both plus and minus branches and producing an inverse relation rather than a function.

Check yourself

Inverse of x−9x-9.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Inverse of x−9x-9.

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Practice 202

Inverse of 4x4x.

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Practice 303

Inverse of x+32\frac{x+3}{2}.

Write a complete attempt before opening the exact answer.

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Complete a substantive attempt before revealing the server-held answer.

Practice 404

Inverse versus reciprocal.

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Practice 505

Explain why this conclusion is valid: f−1(x)=x+73f^{-1}(x)=\frac{x+7}{3}. Use the foundation problem as evidence: Find inverse of 3x−73x-7.

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Practice 606

Solve the representation example, then name the feature of constructing inverse functions that it illustrates: Inverse of4x4x

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is keeping both plus and minus branches and producing an inverse relation rather than a function.

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Practice 808

Connect two representations for this example: Find inverse of 3x−73x-7. 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 of x+32\frac{x+3}{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 constructing inverse functions, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Graphs and verification of 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.