Explanation:
The initial function is:
[tex]f(x)=\frac{1}{9}x-2[/tex]So, to find the inverse function f^(-1)(x), we first need to replace f(x) by y as:
[tex]y=\frac{1}{9}x-2[/tex]Then, we need to solve the equation for x as follows:
[tex]\begin{gathered} y+2=\frac{1}{9}x-2+2 \\ y+2=\frac{1}{9}x \\ 9\cdot(y+2)=9\cdot\frac{1}{9}x \\ 9(y+2)=x \\ 9y+18=x \end{gathered}[/tex]Now, we need to interchange x and y as:
[tex]\begin{gathered} 9x+18=y \\ or \\ y=9x+18 \end{gathered}[/tex]