3. Let $f(x)$ be a function defined on $\mathbf{R}$. If $f(x) + x^2$ is an odd function, and $f(x) + 2^x$ is an even function, then the value of $f(1)$ is $\qquad$ .
Answer: $-\frac{7}{4}$.
Solution: From the given conditions, we know that $f(1)+1=-\left(f(-1)+(-1)^{2}\right)=-f(-1)-1, f(1)+2=f(-1)+\frac{1}{2}$. Adding the two equations to eliminate $f(-1)$, we get $2 f(1)+3=-\frac{1}{2}$, which means $f(1)=-\frac{7}{4}$.