You can see graph of the function [tex]y=3 x^{2} -5[/tex] in the first picture. Now to find its inverse, we are going to switch [tex]x[/tex] and [tex]y[/tex] in our function, and then solve for [tex]y[/tex]
[tex]y=3 x^{2} -5[/tex]
[tex]x=3y^{2} -5[/tex]
[tex]x+5=3y^{2}[/tex]
[tex]y^{2}= \frac{x+5}{3} [/tex]
[tex]y=+or- \sqrt{ \frac{x+5}{3} } [/tex]
Remember that every time that you take a square root, you will have tow results: one positive and one negative. Therefore, our function [tex]y=3 x^{2} -5[/tex] will have tow inverses: [tex]y= \sqrt{ \frac{x+5}{3} } [/tex] and [tex]y=- \sqrt{ \frac{x+5}{3} } [/tex].
In the first picture you can see the graph of the original function; in the second one the graphs of its inverses, and in the third one all the graphs together. Notice that our original function is a quadratic function, and quadratic functions don't have inverse functions unless their domains are restricted.
