Question

Find the inverse of f(x)=x^2-3 and state whether the inverse is a function or not.

Answer 1

as you already know, to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solving for "y", let's do so.

`\stackrel{f(x)}{y}~~ = ~~x^2 - 3\hspace{5em}\stackrel{\textit{quick switcheroo}}{x~~ = ~~y^2 - 3} \\\\\\ x+3=y^2\implies √(x+3)\stackrel{ f^(-1)(x) }{=y}`

now, is it a function or not?

well, is really the graph of a horizontal parabola, that has symmetry over the x-axis, so since we have a mirror image of it above and below the x-axis, it can't be a function, because a function will need to pass the vertical line test, this one doesn't pass it.

Answer 2

Since f(x) = x^2 - 3 is not a one-to-one function, it does not have an inverse. The graph of this function does not pass the horizontal line test.

If you tried to algebraically find the inverse, the work would look like this:

`\begin{aligned}y &= x^2 - 3\\[0.5em]x &= y^2 - 3 ~~~~~~ \text{swapping $x$ and $y$}\\[0.5em]x+3 &= y^2 \\[0.5em]\pm√(x+3) &= y~~~~~~\text{using the square root property}\end{aligned}`

Because the original function is not one-to-one, when you try to find the inverse, there is no unique solution, meaning the inverse equation is not a function. It won't pass the vertical line test.