[tex]\large\boxed{y=\pm\sqrt{\frac{x+3}{5}}}[/tex]
To find the inverse, swap the [tex]y[/tex] (or [tex]f(x)[/tex] in this case) and the [tex]x[/tex], then solve for [tex]y[/tex].
[tex]\begin{aligned}y&=5x^2-3\\x&=5y^2-3\\x+3&=5y^2-3+3\\x+3&=5y^2\\\frac{x+3}{5}&=\frac{5y^2}{5}\\\frac{x+3}{5}&=y^2\\\pm\sqrt{\frac{x+3}{5}}&=\sqrt{y^2}\\\pm\sqrt{\frac{x+3}{5}}&=y\end{aligned}[/tex]
