Given:
There are given that the function to find the inverse is:
[tex]f(x)=9+\sqrt[]{4x-4}[/tex]Explanation:
To find the inverse of the given function, first, we need to exchange f(x) into y, then exchange the variable which means change y to x and x to. Then, find the value of y.
Now,
Step 1:
Exchange f(x) into y:
[tex]\begin{gathered} f(x)=9+\sqrt[]{4x-4} \\ y=9+\sqrt[]{4x-4} \end{gathered}[/tex]Step 2:
Exchange the variables which means exchange y into x and x into y:
[tex]\begin{gathered} y=9+\sqrt[]{4x-4} \\ x=9+\sqrt[]{4y-4} \end{gathered}[/tex]Step 3:
Solve the above function for the value of y:
So,
[tex]\begin{gathered} x=9+\sqrt[]{4y-4} \\ x=9+2\sqrt[]{(y-1)} \\ 2\sqrt[]{(y-1)}=x-9 \\ \sqrt[]{(y-1)}=\frac{x-9}{2} \end{gathered}[/tex]Then,
[tex]\begin{gathered} \sqrt[]{(y-1)}=\frac{x-9}{2} \\ (y-1)^{\frac{1}{2}}=\frac{x-9}{2} \\ y-1=(\frac{x-9}{2})^2 \\ y=(\frac{x-9}{2})^2+1 \end{gathered}[/tex]Then,
[tex]\begin{gathered} y=(\frac{x-9}{2})^2+1 \\ y=\frac{x^2-18x+81}{4}^{}+1 \\ y=\frac{x^2-18x+81+4}{4} \\ f^{-1}(x)=\frac{x^2-18x+85}{4} \end{gathered}[/tex]Final answer:
Hence, the numerator of the inverse function and the denominator of the inverse function are shown below:
[tex]\begin{gathered} \text{The numerator of f}^{-1}(x)\text{ = }x^2-18x+85 \\ \text{The denominator of f}^{-1}(x)\text{ = 4} \end{gathered}[/tex]