I have $\displaystyle \mathbb{E}[X]=\mu$
$\displaystyle
f(x) = \left\{
\begin{array}{lr}
2x^2 + x, & x \leq \mu\\
x^2 + 3x + 1, & x \geq \mu
\end{array}
\right.
$
How do I find $\displaystyle \mathbb{E}[f(X)]$?
You don't have enough information to do any thing but write the expectation as an integral:
$\displaystyle
E(f(X))=\int_{-\infty}^{\infty} f(x) p(x)~dx=\int_{-\infty}^{\mu}(2x^2 + x) p(x)~dx +
\int_{\mu}^{\infty}(x^2 + 3x + 1) p(x)~dx
$
where $\displaystyle p(x)$ the density of $\displaystyle X$.
CB