{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
{{ searchError }}
search
{{ courseTrack.displayTitle }}
{{ printedBook.courseTrack.name }} {{ printedBook.name }}

Manipulating Radical Functions 1.10 - Solution

To find the inverse of $f(x),$ we first need to replace $f(x)$ with $y.$ $\begin{gathered} f(x)=4-2\sqrt{x} \quad \rightarrow \quad y=4-2\sqrt{x} \end{gathered}$ Next step is to switch $x$ and $y$ in the function rule. $y=4-2\sqrt{x} \quad \stackrel{\text{switch}}{\longrightarrow} \quad x=4-2\sqrt{y}$ Now we need to solve for $y.$ The resulting equation will be the inverse of the given function.
$x=4-2\sqrt{y}$
Solve for $y$
$x-4=\text{-} 2\sqrt{y}$
$\dfrac{x-4}{\text{-} 2}=\sqrt{y}$
$\left(\dfrac{x-4}{\text{-} 2}\right)^2=\left(\sqrt{y}\right)^2$
$\dfrac{(x-4)^2}{(\text{-} 2)^2}=\left(\sqrt{y}\right)^2$
$\dfrac{(x-4)^2}{4}=\left(\sqrt{y}\right)^2$
$\dfrac{1}{4}(x-4)^2=\left(\sqrt{y}\right)^2$
$\dfrac{1}{4}(x-4)^2=y$
$y=\dfrac{1}{4}(x-4)^2$
Finally, to indicate that this is the inverse of $f(x),$ we will replace $y$ with $f^{\text{-} 1}(x).$ $\begin{gathered} y=\dfrac{1}{4}(x-4)^2 \quad \rightarrow \quad {\color{#0000FF}{f^{\text{-} 1}(x)}}=\dfrac{1}{4}(x-4)^2 \end{gathered}$