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

Dan Finlay

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DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

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Self-Inverse Functions

What are self-inverse functions?

  • A function space f left parenthesis x right parenthesis is called self-inverse if

    • left parenthesis f ring operator f right parenthesis left parenthesis x right parenthesis equals x for all values of x

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

  • Examples of self-inverse functions include:

    • Identity function: space f left parenthesis x right parenthesis equals x

    • Reciprocal function: space f left parenthesis x right parenthesis equals 1 over x

    • Linear functions with a gradient of -1: space f left parenthesis x right parenthesis equals negative x plus c

What are the symmetries of graphs of self-inverse functions?

  • The graph of a self-inverse function has reflective symmetry

    • The graph is unchanged by a reflection in the line y = x

2-3-3-ib-aa-hl-self-inverse-functions

Examiner Tips and Tricks

  • If your expression for  f to the power of negative 1 end exponent left parenthesis x right parenthesis  is not the same as the expression for  f left parenthesis x right parenthesis  you can check their equivalence by plotting both on your GDC

    • If equivalent the graphs will sit on top of one another and appear as one 

    • This will indicate if you have made an error in your algebra, before trying to simplify/rewrite to make the two expressions identical

  • It is sometimes easier to consider self inverse functions geometrically rather than algebraically

Worked Example

Use algebra to show the function defined by space f open parentheses x close parentheses equals fraction numerator 7 x minus 5 over denominator x minus 7 end fraction comma blank x not equal to 7 is self-inverse.

2-3-3-ib-aa-hl-self-inverse-functions-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.