Answer :
[tex]g^{-1}(x)=\pm(\sqrt[]{x-36+6x})[/tex]
Explanation
[tex]g(x)=x^2-6x+36[/tex]The inverse function returns the original value for which a function gave the output
Step 1
swap x and y and solve for y
so
[tex]\begin{gathered} y=x^2-6x+36 \\ \text{swap x and y } \\ x=y^2-6x+36 \\ \end{gathered}[/tex]now, isolate y
[tex]\begin{gathered} x=y^2-6x+36 \\ \text{subtract 36 in both sides} \\ x-36=y^2-6x+36-36 \\ x-36=y^2-6x \\ \text{add 6x in both sides} \\ x-36+6x=y^2-6x+6x \\ x-36+6x=y^2 \\ get\text{ the square root in both sides} \\ \sqrt{x-36+6x}=\sqrt{y^2} \\ \pm(\sqrt[]{x-36+6x})=y \\ y=\pm(\sqrt[]{x-36+6x}) \end{gathered}[/tex]note, there is a negative answer , it is because when gettin the square we get y , so
the answer is
[tex]g^{-1}(x)=\pm(\sqrt[]{x-36+6x})[/tex]I hope this helps you