Expected value of a function of a finite moment random variable

Question

Say we have a random variable $X$ such that $\mathbb{E}[X]<\infty$. We also have a function $f:\mathbb{R}\to\mathbb{R}$. Is it necessarily true that $\mathbb{E}[f(X)]<\infty$?

Answer 1

Let $X$ have the density $\displaystyle f(x)=\frac{2}{x^3}\mathbf{1}_{1<x<\infty}$.

Then $\mathbb{E}(X)$ exists, but $\mathbb{E}(X^2)$ and hence $\mathbb{V}(X)$ do not exist.

Answer 2

No.

E.g. there are lots of random variables with $\mathsf{E}|X|<\infty$ and $\mathsf{E}X^2=+\infty$.