Inverse Functions

Explore one-to-one requirement, swapping inputs and outputs, inverse graphs.

Lesson 2 of 53 parts Practice Flashcards

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One-to-One Requirement

Objective: Explain one-to-one requirement and use it to solve or interpret a problem.

A function has an inverse function on its domain only when each output comes from one input. Restricting a quadratic branch can meet this condition. Every horizontal line meets the graph at most once.

Worked examples

Try itInverse Functions — practice 1.1: How many inputs give output 16 under f(x)=2x+6?Show answer

Step 1: The slope 2 is nonzero. Step 2: Solving 2x+6=16 gives x=5. Step 3: Exactly one input produces that output.

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One-to-One Requirement

Objective: Explain one-to-one requirement and use it to solve or interpret a problem.

Worked examples

Try itInverse Functions — practice 1.1: How many inputs give output 16 under f(x)=2x+6?Show answer

Step 1: The slope 2 is nonzero. Step 2: Solving 2x+6=16 gives x=5. Step 3: Exactly one input produces that output.

Swapping Inputs and Outputs

Objective: Explain swapping inputs and outputs and use it to solve or interpret a problem.

To find an inverse formula, set y=f(x), swap x and y, solve for y, and exchange domain and range. Check compositions where defined. Original outputs become inverse inputs.

Worked examples

Try itInverse Functions — practice 2.1: If f(x)=3x+7, find f⁻¹(31).Show answer

Step 1: Set 3x+7=31. Step 2: Subtract 7 and divide by 3. Step 3: The inverse output is 8.

Inverse Graphs

Objective: Explain inverse graphs and use it to solve or interpret a problem.

Points (u,v) of a function become (v,u) on its inverse; their graphs reflect across y=x on corresponding restricted domains. No, it repeats outputs unless its domain is restricted.

Worked examples

Try itInverse Functions — practice 3.1: If f(x)=4x+3, find f⁻¹(47).Show answer

Step 1: Set 4x+3=47. Step 2: Subtract 3 and divide by 4. Step 3: The inverse output is 11.

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