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Inverse Functions (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

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Inverse Functions

What is an inverse function?

  • Only one-to-one functions have inverses

  • A function has an inverse if its graph passes the horizontal line test

    • Any horizontal line will intersect with the graph at most once

  • The identity function id maps each value to itself

    • id left parenthesis x right parenthesis equals x

  • If space f ring operator g and space g ring operator f have the same effect as the identity function then space f and space g are inverses

  • Given a function space f left parenthesis x right parenthesis we denote the inverse function as space f to the power of negative 1 end exponent left parenthesis x right parenthesis

  • An inverse function reverses the effect of a function

    • space f left parenthesis 2 right parenthesis equals 5 means space f to the power of negative 1 end exponent left parenthesis 5 right parenthesis equals 2

  • Inverse functions are used to solve equations

    • The solution of space f left parenthesis x right parenthesis equals 5 is x equals f to the power of negative 1 end exponent left parenthesis 5 right parenthesis

  • A composite function made of space f and space f to the power of negative 1 end exponent has the same effect as the identity function

    • left parenthesis f ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals left parenthesis f to the power of negative 1 end exponent ring operator f right parenthesis left parenthesis x right parenthesis equals x

Language of Functions Notes Diagram 9

What are the connections between a function and its inverse function?

  • The domain of a function becomes the range of its inverse

  • The range of a function becomes the domain of its inverse

  • The graph of space y equals f to the power of negative 1 end exponent left parenthesis x right parenthesis is a reflection of the graph space y equals f left parenthesis x right parenthesis in the line space y equals x

    • Therefore solutions to space f left parenthesis x right parenthesis equals x or space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals x will also be solutions to space f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left parenthesis x right parenthesis

      • There could be other solutions to space f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left parenthesis x right parenthesis that don't lie on the line space y equals x

Inverse Functions Notes Diagram 2

How do I find the inverse of a function?

  • STEP 1: Swap the x and y in space y equals f left parenthesis x right parenthesis

    • If space y equals f to the power of negative 1 end exponent left parenthesis x right parenthesis then x equals f left parenthesis y right parenthesis

  • STEP 2: Rearrange x equals f left parenthesis y right parenthesis to make space y the subject

  • Note this can be done in any order

    • Rearrange space y equals f left parenthesis x right parenthesis to make x the subject

    • Swap x and space y

Examiner Tips and Tricks

  • Remember that an inverse function is a reflection of the original function in the line y equals x

    • Use your GDC to plot the function and its inverse on the same graph to visually check this

  • space f to the power of negative 1 end exponent left parenthesis x right parenthesis  is not the same as  fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction

Worked Example

For the function space f left parenthesis x right parenthesis equals fraction numerator 2 x over denominator x minus 1 end fraction comma space x greater than 1:

a) Find the inverse of space f left parenthesis x right parenthesis.

2-3-2-ib-aa-sl-inverse-functions-a-we-solution

b) Find the domain of space f to the power of negative 1 end exponent left parenthesis x right parenthesis.

2-3-2-ib-aa-sl-inverse-functions-b-we-solution

c) Find the value of k such that f left parenthesis k right parenthesis equals 6.

2-3-2-ib-aa-sl-inverse-functions-c-we-solution

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    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.