To determine the inverse of a function:
[tex]y=\frac{\sqrt[]{2x-3}}{3}[/tex]
A function g is the inverse function of f, if for y = f(x) , x = g(y)
[tex]\begin{gathered} y=\frac{\sqrt[]{2x-3}}{3} \\ 3y=\sqrt[]{2x-3} \\ \text{square both side of the equation} \\ (3y)^2=(\sqrt[]{2x-3)^2} \\ 9y^2=2x-3 \\ 9y^2+3=2x \\ \text{divide both side by 2} \\ \frac{2x}{2}=\frac{9y^2+3}{2} \\ x=\frac{9y^2+3}{2} \\ \text{Then replacing y with x} \\ y=\frac{9x^2+3}{2} \end{gathered}[/tex]
Therefore inverse function for y = f(x) , x = g(y) where x ≥ 0
[tex]y=\frac{9x^2+3}{2},\text{ where x }\ge0[/tex]
Hence the correct answer is Option C
