Question

What is the inverse function of y = (x-4)^2+2

Answer 1

One way to find the inverse of a function is by first swapping x and y, then solving for y, like this:

$\begin{gathered} y=(x-4)^2+2\text{ }\Rightarrow x=(y-4)^2+2 \\ \end{gathered}$

Now, let's solve for y, like this:

$\begin{gathered} x=(y-4)^2+2 \\ x-2=(y-4)^2+2-2 \\ (y-4)^2=x-2 \\ \sqrt[]{\mleft(y-4\mright)^2}=\sqrt[]{x-2} \\ y-4=\sqrt[]{x-2} \\ y-4+4=\sqrt[]{x-2}+4 \\ y=\sqrt[]{x-2}+4 \end{gathered}$

Then, the inverse function of y = (x-4)^2+2​ is:

$y=\sqrt[]{x-2}+4$