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

Self-inverse functions

What are self-inverse functions?

• A function $f\left(x\right)$ is called self-inverse if it is equal to its inverse function $f^{-1}\left(x\right)$

  • $f\left(x\right)=f^{-1}\left(x\right)$

• Applying $f$ to both sides of $f\left(x\right)=f^{-1}\left(x\right)$ gives

  • $(f\circ f)\left(x\right)=ff\left(x\right)=x$

    • as $f$ of $f^{-1}$ cancels out on the right-hand side

What are examples of self-inverse functions?

• Examples of common self-inverse functions include

  • the identity function $f\left(x\right)=x$

  • the reciprocal function $f\left(x\right)=\frac{1}{x}$

  • Linear functions with a gradient of -1

    • e.g. $f\left(x\right)=8-x$

What do the graphs of self-inverse functions look like?

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

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