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A.4(C)·Algebra II·Function TransformationsSupporting

Functions and Their Inverses

Describe and analyze the relationship between a function and its inverse.

Students find inverses by exchanging inputs and outputs and identify linear functions whose inverses have the same rule.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

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Learn the concept

Reverse the mapping

An inverse maps each output back to its original input. If (a,b)(a,b) lies on a one-to-one function, (b,a)(b,a) lies on its inverse.

Swap and solve

Write y=f(x)y=f(x), interchange x and y, and solve for y. For a nonconstant linear function this produces another linear function defined for all real inputs.

Recognize self-inverse rules

A function is self-inverse when its inverse has exactly the same rule on the same domain. For example, any f(x)=c−xf(x)=c-x satisfies f(f(x))=c−(c−x)=xf(f(x))=c-(c-x)=x. Its graph is symmetric about y=xy=x.

Worked example

Which function equals its inverse: p(x)=10−xp(x)=10-x or q(x)=x−6q(x)=x-6?

  1. For p, swap: x=10−yx=10-y, so y=10−xy=10-x.
  2. For q, swap: x=y−6x=y-6, so y=x+6y=x+6, not x−6x-6.
  3. Confirm p(p(x))=10−(10−x)=xp(p(x))=10-(10-x)=x.

Only p(x)=10−xp(x)=10-x equals its inverse.

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Functions and Their Inverses

UT IPSI · Kid2Kid / Texas Education Agency

Exchange inputs and outputs and compare a function with its inverse.

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TEKSForge practice · unscored

  1. 1. Find the inverse of f(x)=8−x2f(x)=\frac{8-x}{2}.

    Show answer and reasoning

    f−1(x)=8−2xf^{-1}(x)=8-2x

    Swap to get x=(8−y)/2x=(8-y)/2; then 2x=8−y2x=8-y, so y=8−2xy=8-2x. This is not the original rule.

  2. 2. Is g(x)=−3−xg(x)=-3-x self-inverse?

    Show answer and reasoning

    Yes.

    g(g(x))=−3−(−3−x)=xg(g(x))=-3-(-3-x)=x for every real input.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 41 · 1 point

Which function is equivalent to its inverse?

  1. f(x)=2x−12f(x) = \frac{2x - 1}{2}
  2. g(x)=3−x3g(x) = \frac{3 - x}{3}
  3. h(x)=7−xh(x) = 7 - x
  4. j(x)=x−4j(x) = x - 4
Show STAAR answer and explanation

h(x)=7−xh(x) = 7 - x

For h, swap x and y and solve: x=7−yx=7-y gives y=7−xy=7-x, the same rule. Its graph is symmetric about the line y=xy=x.

f(x)=2x−12f(x) = \frac{2x - 1}{2}: f(x)=x−12f(x)=x-\frac12, so its inverse is x+12x+\frac12.

g(x)=3−x3g(x) = \frac{3 - x}{3}: Solving x=3−y3x=\frac{3-y}{3} gives the inverse 3−3x3-3x.

j(x)=x−4j(x) = x - 4: The inverse of x−4x-4 is x+4x+4.

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Learning objectives

By the end of the lesson, students can:

  • Find an inverse by swapping x and y and solving.
  • Compare a function rule with its inverse.
  • Verify a self-inverse function through composition.

Instructional sequence

  1. 1

    Engage

    5 min

    Swap ordered pairs in a small table.

  2. 2

    Direct instruction

    10 min

    Model finding an inverse of a linear function.

  3. 3

    Guided practice

    13 min

    Compare inverse rules to identify self-inverse functions.

  4. 4

    Compare and verify

    15 min

    Confirm algebra with composition and reflected points.

  5. 5

    Closure

    7 min

    Explain why one rule equals its inverse on the exit ticket.

Misconceptions to address

“The inverse is the reciprocal.”

Teacher move: An inverse reverses the input-output process; a reciprocal changes a value multiplicatively.

“Every decreasing line equals its inverse.”

Teacher move: The reflected graph must coincide with the original; test the inverse rule rather than direction alone.