Inverse Functions

Starting point: what you already know

• A function gives exactly one output for each input. The vertical line test: if any vertical line cuts the graph more than once, it is not a function.
• The domain is the set of allowed inputs. The range is the set of outputs.
• Solving for a variable: from $y=2x+3$ you can get $x=\frac{y-3}{2}$.
• From Mathematics Advanced: $y=e^x$ and $y=\ln x$ undo each other, so $\ln (e^x)=x$.

Undoing a function

The inverse function $f^{-1}$ reverses what $f$ does. If $f$ sends 5 to 13, then $f^{-1}$ sends 13 back to 5.

$$f(a)=b⟺f^{-1}(b)=a$$

Which functions can be undone?

Think of $f(x)=x^2$. Both 3 and $-3$ give 9. If you are told the answer is 9, you cannot tell which input it came from. So $x^2$ cannot be undone on all real numbers.

A function is one-to-one if every output comes from exactly one input. Only one-to-one functions have an inverse.

Horizontal line test: if every horizontal line cuts the graph at most once, the function is one-to-one.

Finding the inverse

1. Write $y=f(x)$.
2. Swap $x$ and $y$.
3. Make $y$ the subject. That $y$ is $f^{-1}(x)$.

The mirror in y = x

Swapping $x$ and $y$ turns the point $(a,b)$ into $(b,a)$. On a graph, that is a reflection in the line $y=x$. So:

The graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ in the line $y=x$.

Because the coordinates swap, the domain and range swap too:

• domain of $f^{-1}$ = range of $f$
• range of $f^{-1}$ = domain of $f$

Quick summary

• $f^{-1}$ undoes $f$: if $f(a)=b$ then $f^{-1}(b)=a$.
• Only one-to-one functions have inverses. Use the horizontal line test.
• To find $f^{-1}$, swap $x$ and $y$, then make $y$ the subject.
• The graph of $f^{-1}$ is the reflection of $f$ in $y=x$. Domain and range swap.
• Restrict the domain to make a function one-to-one.

Quiz