Give example of convex and not convex

Question

I'm looking for a little input on this question.

Let $f\colon X\rightarrow (-\infty,+\infty)$ be convex and let $I$ be an interval in $\mathbb{R}$ (bounded or unbounded). Further, recall that $f^{-1}(I)= \{\,x\in X : f(x)\in I\,\}$ by definition. Provide examples where:

1. $f^{-1}(I)$ is convex
2. $f^{-1}(I)$ is non convex.

I am wondering if this question is simply asking me to provide an example of a function such that the inverse can be convex in some cases and non convex in others. For example, consider f(x)=x. The inverse is $f^{-1}(x)=\frac{1}{x}$, which in convex on the interval $[0,+\infty)$ and non convex on the interval $(-\infty,0]$.

any feed back would be wonderful.

Thank you.

Answer 1

Hint: Try $f\colon\mathbb R\to\mathbb R$, $x\mapsto x^2$ with $I=[-1,1]$ and with $I=[1,4]$.