Find and graph $f^{-1}$

Question

We have $f: \mathbb{R}^2 \to \mathbb{R}, \ f(x,y)=x^2y, \ A = (-3,1] \times [-2,2], \ B = [-1,2)$. We want to find $f[A]$ and find and graph $f^{-1}[B]$.

$f[A] = (-18,18)$ but I get stuck on finding $f^{-1}$. How might I approach this?

Answer 1

Look at the graphs of:

• $ x^2y = -1$
• $ x^2y = 2$

Solve for $y = \dots$, and plot them both. Those are the borders of your interval in the preimage, and it should be quite easy to decide which areas belong to the preimage and which not.

Answer 2

Hint for the first.

$$f(A)=f([0,3)×[-2,2])=(-18,18)$$