Question

Graph y = -­4x2 ­- 2 and its inverse.

Answer 1

Be careful with finding its inverse! An inverse only exists for one-to-one functions so we need to restrict the domain such that the quadratic is one-to-one. For brevity, we'll take the domain: $x \geq 0$ but you could just as easily take $x \leq 0$.

Being that the quadratic is now one-to-one, we'll try to find its inverse.

By definition, an inverse function is the function flipped across the line y = x. So being that, we need to interchange the x and y coordinates around and then solve for y.

This gives us:

$x = -4y^2 - 2 \\ x + 2 = -4y^2 \\ -\frac{1}{4}\left(x + 2\right) = y^2 \\ y = \sqrt{-\frac{1}{4}\left(x + 2\right)}$

Now, this function is defined for when $y \geq 0$ and $x \leq -2$.

So I'll end with this, what is the relationship between the domain/range of the original function and the domain/range of the inverse?