To find the inverse of a function, you have to start by replacing the f(x) by y:
[tex]\begin{gathered} f(x)=-x^2+6 \\ y=-x^2+6 \end{gathered}[/tex]
Now switch the x and y:
[tex]x=-y^2+6[/tex]
Solve for y:
[tex]\begin{gathered} \text{Subtract 6 from both sides} \\ x-6=-y^2+6-6 \\ x-6=-y^2 \\ \text{ Multiply both sides by -1} \\ -1(x-6)=-1(-y^2) \\ 6-x=y^2 \\ \text{Apply square root to both sides} \\ \sqrt[]{6-x}=\sqrt[]{y^2} \\ \sqrt[]{6-x}=y \\ \text{Then,} \\ y=\sqrt[]{6-x} \end{gathered}[/tex]
Finally, replace y by f^-1(x):
[tex]f^{-1}(x)=\sqrt[]{6-x}[/tex]
The graph of the main function is in red, and the graph of the inverse function is in blue:
As you can see, the graph of the inverse function (blue) is the graph of the main function (red) reflected about the diagonal line y=x (orange). Then, the inverse relationship is verified.
