Question

Graph y = 3x to the second power - 5 and its inverse

Answer 1

You can see graph of the function $y=3 x^{2} -5$ in the first picture. Now to find its inverse, we are going to switch $x$ and $y$ in our function, and then solve for $y$
$y=3 x^{2} -5$
$x=3y^{2} -5$
$x+5=3y^{2}$
$y^{2}= \frac{x+5}{3}$
$y=+or- \sqrt{ \frac{x+5}{3} }$
Remember that every time that you take a square root, you will have tow results: one positive and one negative. Therefore, our function $y=3 x^{2} -5$ will have tow inverses: $y= \sqrt{ \frac{x+5}{3} }$ and $y=- \sqrt{ \frac{x+5}{3} }$.
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.