MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
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2.8 One-to-One Functions and Their Inverses

Reverse a one-to-one function.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Decide and justify one-to-one behaviorWorked steps · five questions02Read inverse values and reverse input-output rolesWorked steps · five questions03Verify inverses and track their domainsWorked steps · five questions04Invert affine and odd-power rulesWorked steps · five questions05Invert rational functions with excluded valuesWorked steps · five questions06Restrict domains and invert even powers and radicalsWorked steps · five questions07Construct reflected graphs and test complex graphsWorked steps · five questions08Interpret inverse models in contextWorked steps · five questions09Invert piecewise and composite modelsWorked steps · five questions10Prove identity properties and solve functional equationsWorked steps · five questions
Lesson brief and rules

2.8 One-to-One Functions and Their Inverses

An inverse reverses a function’s input and output. For a nonzero slope, solve y=ax+by = ax + b for xx, then interchange the variable names.

Remember
The notation f⁻¹ denotes an inverse function, not the reciprocal 1/f.
Worked example 11
01 / Prove directly from equal outputs that f(x)=−2x+5f(x)=-2x+5 is one-to-one on ℝ. Do not use the vertical-line test as your argument. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / Assume −2u+5=−2v+5-2u+5=-2v+5.
03 / Subtract 55, then divide by the nonzero coefficient −2-2.
04 / This gives u=vu=v, exactly the one-to-one condition.
Worked example 2 · Decide and justify one-to-one behavior

Is f(x)=f(x)=(x−3)2\left(x - 3\right)^{2} one-to-one on ℝ? Give two distinct inputs with equal outputs to support your answer. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

The inputs 22 and 44 differ.
Both are one unit from 33, so both outputs equal 11.
One repeated output at distinct inputs disproves injectivity.
Written-work review:

Read the graphs

Decide and justify one-to-one behavior

Use the horizontal-line test on the supplied graph to decide whether ff is one-to-one. Explain your evidence.
Function graphThe graph is made of straight segments joining the marked vertices.-4-3-2-101234-3-2-112345xyf
The graph is made of straight segments joining the marked vertices.

Trace horizontal, not vertical, lines through the curve.
Each segment rises strictly, so no height repeats.
Yes; every horizontal line meets the graph at most once.

Answer: Yes; every horizontal line meets the graph at most once.

Read inverse values and reverse input-output roles

Read f−1(4)f^{-1}(4) from the graph by starting at output 44.
Function graphA strictly increasing piecewise-linear function with included endpoints.-3-2-101234567-2-11234567xyf
A strictly increasing piecewise-linear function with included endpoints.

At height 44, move horizontally to the graph.
The corresponding input is 22; therefore f−1(4)=2f^{-1}(4)=2.

Answer: 22

Construct reflected graphs and test complex graphs

Sketch the inverse of the supplied function by reflection. Label every corner and both endpoints; do not replace the drawing with an equation alone.
Function graphThe entire graph is the solid polyline between the four marked vertices; endpoints included.-3-2-101234567-3-2-11234567xyf
The entire graph is the solid polyline between the four marked vertices; endpoints included.
Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Swap the coordinates of every displayed vertex.
The new vertices are [(−1,−2),(1,0),(3,2),(5,4)][(-1, -2), (1, 0), (3, 2), (5, 4)].
Straight segments reflect to straight segments, and included endpoints remain included.
Written-work review:

  • Every reflected vertex correct.
  • Segments connect in increasing inverse-input order.
  • Both endpoints and the line of reflection shown.

Answer: Connect (−1,−2),(1,0),(3,2),(5,4)(-1, -2), (1, 0), (3, 2), (5, 4) in that order with straight segments.

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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