Inverse Functions (College Board AP® Precalculus): Revision Note

Inverse functions

What is an inverse function?

• An inverse function is a function that reverses the mapping of an original function

  • For the function $f$, the inverse function is notated by ${\mathit{f}}^{\mathbf{-}\mathbf{1}}$

• If the original function takes input $a$ and produces output $b$

  • then the inverse function takes $b$ as input and returns output $a$

    • I.e. if $f\left(a\right)=b$

      • then $f^{-1}\left(b\right)=a$

• You can think of this in terms of input-output pairs

  • If $f$ contains the pair $(a,b)$

    • then $f^{-1}$ contains the pair $(b,a)$

• This means you can find values of an inverse function

  • by reading the original function backwards

• From a table

  • If you know that $f\left(3\right)=7$, then $f^{-1}\left(7\right)=3$

    • Look for 7 in the output row, then read the corresponding input

• From a graph

  • If the point $(3,7)$ is on the graph of $f$, then $f^{-1}\left(7\right)=3$

    • Find 7 on the $y$-axis, read across to the curve, and then down to the $x$-axis

When does a function have an inverse?

• A function $f$ has an inverse function (i.e. it is invertible) on a given domain

  • if every output value comes from exactly one input value

    • In other words, if no two different inputs produce the same output

  • Such a function, where no two different inputs produce the same output, is often referred to as a one-to-one function

    • A function must be one-to-one to be invertible

• This invertibility requirement comes because an inverse function must be a function

  • And a function can only give one output for each input

• If a function is not invertible on its full domain, the domain can sometimes be restricted to a smaller interval where it is invertible

  • There is often more than one way to restrict the domain to achieve this

• E.g. the function $f\left(x\right)=x^2$ is not invertible on all real numbers

  • because both $f\left(2\right)=4$ and $f(-2)=4$

    • Two different inputs give the same output

• But if the domain is restricted to $x\ge 0$

  • then each output comes from a unique input, and the inverse exists

    • In that case $f^{-1}\left(x\right)=\sqrt{x}$

How do you determine if a function is invertible from a table or graph?

• From a table

  • Check whether any output value appears more than once

    • If it does, the function is not invertible (on the given domain)

    • because multiple inputs map to the same output

• From a graph

  • You can use the 'horizontal line test'

  • Check whether any horizontal line crosses the graph more than once

    • If it does, there are multiple inputs producing the same output,

    • so the function is not invertible

• If the function is always increasing or always decreasing on its domain

  • then it is automatically invertible

    • because outputs always changing in only one direction

    • means no two inputs can share the same output