 ## FG3 Inverse notation If f-1(x) is the inverse function of a one-to-one function f(x) then f-1(x) is the set of ordered pairs obtained by interchanging the first and second elements in each ordered pair.

### Definition of an Inverse Function

If $$f^{-1}(x)$$ is the inverse funtion of a one-to-one function $$f(x)$$ then $$f^{-1}(x)$$ is the set of all ordered pairs obtained by interchanging the first and second elements in each ordered pair.

So if $$(a,b)\in f$$ then $$(b,a)\in f^{-1}$$ and if $$f(a)=b$$ then $$f^{-1}(b)=a$$

The domain of $$f$$ is the range of $$f^{-1}$$ and the range of $$f$$ is the domain of $$f^{-1}$$

For example the function $$f:R\rightarrow R$$, defined by $$y=f(x)=\frac{x-1}{2}$$ has an inverse function with the rule $$y=2x+1$$ .

So $$(3,1)$$ belongs to $$f$$ and $$(1,3)$$ belongs to $$f^{-1}$$ and $$(-7,-4)$$ belongs to $$f$$ and $$(-4,-7)$$ belongs to $$f^{-1}$$ .

### Graph of an Inverse Function

The graphs of any one-to-one function, $$f$$ , and its inverse, $$f^{-1}$$ , are symmetric about the line $$y=x$$ . ### Finding an Inverse Function for $$y=f(x)$$

To obtain the rule for an inverse function, swap the $$x$$ and $$y$$ coordinates in $$f$$ and rearrange to express $$y$$ in terms of $$x$$ .

#### Example

Find the inverse function of $$f$$ where $$f(x)=2-3x$$

\begin{align*} y & =2-3x\\ x & =2-3y\text{ \quad (swap the x\text{ and y) } }\\ x-2 & =-3y\text{ \quad (rearrange to make y the subject)}\\ -x+2 & =3y\\ \frac{-x+2}{3} & =y\\ f^{-1}(x) & =\frac{-x+2}{3} \end{align*}

#### Exercise

Find the inverse of each of the following one-to-one functions:

1. $$y=x+5$$

2. $$y=4x$$

3. $$y=\frac{2x+1}{3}$$

4. $$y=\sqrt{2x-1}$$ , $$x\geq\frac{1}{2}$$

1. $$f^{-1}(x)=x-5$$
2. $$f^{-1}(x)=\frac{x}{4}$$
3. $$f^{-1}(x)=\frac{3x-1}{2}$$
4. $$f^{-1}(x)=\frac{x^{2}+1}{2}$$ , $$x\geq0$$