Inverse Functions (Cambridge (CIE) O Level Additional Maths): Revision Note

Inverse functions

What is an inverse function?

• An inverse function does the exact opposite of the function it came from

  • For example, if the function “doubles the number and adds 1” then its inverse is

  • “subtract 1 and halve the result”

• It is the inverse operations in the reverse order

How do I write inverse functions?

• An inverse function f^-1 can be written as ${\mathrm{f}}^{-1}\left(x\right)=\dots$ or  ${\mathrm{f}}^{-1}:x\mapsto \dots$

  • For example, if $\mathrm{f}\left(x\right)=2x+1$ its inverse can be written as

  • ${\mathrm{f}}^{-1}\left(x\right)=\frac{(x–1)}{2}$  or   ${\mathrm{f}}^{-1}:x\mapsto \frac{(x–1)}{2}$

How do I find an inverse function?

• The easiest way to find an inverse function is to 'cheat' and swap the $x$ and $y$ variables

  • Note that this is a useful method but you MUST remember not to do this in any other circumstances in maths

• STEP 1 Write the function in the form$y=\dots$ e.g.   $y=2x+1$

• STEP 2 Swap the $x$'s and$y$'s to get $x=\dots$ e.g.  $x=2y+1$

• STEP 3 Rearrange the expression to make $y$ the subject again $\begin{array}{rcl}x-1& =& 2y\\ \frac{x-1}{2}& =& y\to y=\frac{x-1}{2}\end{array}$

• STEP 4 Rewrite using the correct notation for an inverse function

  • either as f^-1(x) = … or f^-1 : x ↦ …

  • $y$should not exist in the final answer

    • e.g.  ${\mathrm{f}}^{-1}\left(x\right)=\frac{x-1}{2}$

How does a function relate to its inverse?

• If $\mathrm{f}\left(3\right)=10$ then the input of 3 gives an output of 10

  • The inverse function undoes f(x)

  • An input of 10 into the inverse function gives an output of 3

    • If $\mathrm{f}\left(3\right)=10$ then ${\mathrm{f}}^{-1}\left(10\right)=3$

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

  • If you apply a function to x, then immediately apply its inverse function, you get x

    • Whatever happened to x gets undone

  • f and f^-1 cancel each other out when applied together

• If $\mathrm{f}\left(x\right)=2^x$ and you want to solve ${\mathrm{f}}^{-1}\left(x\right)=5$

  • Finding the inverse function ${\mathrm{f}}^{-1}\left(x\right)$ in this case is tricky (impossible if you haven't studied logarithms)

  • instead, take f of both sides and use that ${\mathrm{ff}}^{-1}$ cancel each other out:

$\begin{array}{rcl}{\mathrm{ff}}^{-1}\left(x\right)& =& \mathrm{f}\left(5\right)\\ x& =& \mathrm{f}\left(5\right)\\ x& =& 2^5=32\end{array}$

What condition is needed for an inverse function to exist?

• For the inverse function to exist, ${\mathrm{f}}^{-1}\left(x\right)$, the original function $\mathrm{f}\left(x\right)$ must be one-to-one

  • Substituting 1 input into $\mathrm{f}$ must give 1 output only

  • Substituting this 1 output into ${\mathrm{f}}^{-1}$ must give back the original input only

    • At no point are more values allowed to be created!

Domain & range of inverse functions

How do I find the domain and range of inverse functions?

• The range of a function will be the domain of its inverse function

• The domain of a function will be the range of its inverse function

Graphs of inverse functions

How are the graphs of a function and its inverse related?

• The graph of an inverse function, $y={\mathrm{f}}^{-1}\left(x\right)$, is a reflection of the graph of the function, $y=\mathrm{f}\left(x\right)$, in the line $y=x$

• Key features of the graph of $y=\mathrm{f}\left(x\right)$ will be reflected, such as

  • $x$ and $y$ axes intercepts

  • turning points

  • asymptotes

How do I sketch the graph of an inverse function?

• STEP 1

  • Sketch the line $y=x$, and if need be, the graph of $y=\mathrm{f}\left(x\right)$

• STEP 2

  • Reflect the graph of $y=\mathrm{f}\left(x\right)$ in the line $y=x$

    • Remember it is a sketch, but the graphs together should look like reflections

  • Consider points where the reflected graph will intersect the $x$ and $y$ axes

    • e.g.  The point $\left(4,0\right)$ will be reflected to the point $\left(0,4\right)$

  • Consider any asymptotes on the graph of $y=\mathrm{f}\left(x\right)$ - these will also be need reflecting

    • e.g.  The asymptote (line) $x=-2$ will be reflected to the line $y=-2$

  • Consider any restrictions on the domain and range of $\mathrm{f}\left(x\right)$

    • e.g.  If the domain is $x>0$ only sketch the graph for positive values of $x$

• STEP 3

  • Label key points on the sketch of $y={\mathrm{f}}^{-1}\left(x\right)$ and state the equations of any asymptotes

• This process works the other way round - the graph of $y=\mathrm{f}\left(x\right)$ can be sketched from the graph of $y={\mathrm{f}}^{-1}\left(x\right)$