Inverse image of the set $[−1, 4)$ under $f : x \mapsto -x^2$

Question

I have an issue with determining the inverse image of a set. I cannot understand anything about it. I've got a simple exercise here, could someone here show me how the inverse image works and more importantly, why?

There is a function $f : \Bbb R \to \Bbb R, f(x) = −x^2$. Find the inverse image of the set $[−1, 4)$.

Answer 1

The inverse image of $A=[-1,4)$ under $f$ is $$ \{x \in \Bbb R \mid -1 ≤ f(x) = -x^2 < 4 \} $$

Then you have to solve w.r.t. $x$ the following inequations : $1≥x^2>-4$.

Here is a plot of $f$ ; the red segment shows the set $A$ :

Answer 2

Hint:

Think about the numbers $x$ such that $-1\leq -x^2<4$.