Manipulating Radical Functions

To find the inverse of

f(x),

we first need to replace

f(x)

with

y.

\begin{gathered} f(x)=\sqrt{x+2} \quad \rightarrow \quad y=\sqrt{x+2} \end{gathered}

Next step is to switch

x

and

y

in the function rule.

y=\sqrt{x+2} \quad \stackrel{\text{switch}}{\longrightarrow} \quad x=\sqrt{y+2}

Now we need to solve for

y.

The resulting equation will be the inverse of the given function.

x=\sqrt{y+2}

Solve for

y

\text{LHS}^{2}=\text{RHS}^{2}

x^2=\left(\sqrt{y+2}\right)^2

\left ( \sqrt{a}\right )^2 = a

x^2=y+2

\text{LHS}-2=\text{RHS}-2

x^2-2=y

RearrangeEqnRearrange equation

y=x^2-2

Finally, to indicate that this is the inverse function of

f(x),

we will replace

y

with

f^{\text{-} 1}(x).

\begin{gathered} y=x^2-2 \quad \rightarrow \quad {\color{#0000FF}{f^{\text{-} 1}(x)}}=x^2-2 \end{gathered}

Manipulating Radical Functions 1.8 - Solution