# Question 9e045

Aug 6, 2017

${f}^{- 1} \left(27\right) = 9$

#### Explanation:

The trick here is to realize that for a given value, the inverse function has its output as the input of the original function for the same value.

This means that if you plug something into the original function (the red apples), you will get an output (the yellow pears). If you then plug that output (the yellow pears) into the inverse function, you will get the something (the red apples) that you plugged into the original function.

The problem wants you to determine the output of the inverse function of $f \left(x\right)$, which we label as ${f}^{- 1} \left(x\right)$, when the input of the inverse function is equal to $\textcolor{b l u e}{27}$.

f^(-1)(color(blue)(27)) = ?

In other words, you need to find the input of $f \left(x\right)$ when its output is equal to $\textcolor{b l u e}{27}$.

So you need

f(?) = 3 * ? = color(blue)(27)

This will give you

? = color(blue)(27)/3 = color(red)(9)#

Therefore, ${f}^{- 1} \left(\textcolor{b l u e}{27}\right) = \textcolor{red}{9}$.

So, if you plug $\textcolor{red}{9}$ into $f \left(x\right)$ you will get an output of $\textcolor{b l u e}{27}$. If you then plug $\textcolor{b l u e}{27}$ into the inverse function, you will get $\textcolor{red}{9}$.

$f \left(\textcolor{red}{9}\right) = 3 \cdot \textcolor{red}{9} = \textcolor{b l u e}{27}$

${f}^{- 1} \left(\textcolor{b l u e}{27}\right) = \frac{\textcolor{b l u e}{27}}{3} = \textcolor{red}{9}$

This shows that the inverse function takes the form

${f}^{- 1} \left(x\right) = \frac{x}{3}$