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5 problems match “functional equations”. Clear

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1. Value of even plus odd function ●●○○○ 20 ptsLet $e(x)$ be an even function and let $o(x)$ be an odd function such that $e(x) + x^2 = o(x)$ for all $x$. Let $f(x) = e(x) + o(x)$. Find $f(2)$.
2. Value of f(3) from symmetry conditions ●●○○○ 20 ptsLet $f(x)$ be a function with domain $\mathbb{R}$ such that $f(x+1) = f(-x-1)$ and $f(2-x) = -f(x)$ for all real $x$. Find the value of $f(3)$.
3. Value missing from range of a Möbius function ●●●○○ 50 ptsThe function $f$ is defined by $f(x)= \frac{ax+b}{cx+d}$, where $a$, $b$, $c$ and $d$ are nonzero real numbers. It has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all $x$ except $-\frac{d}{c}$. Find the unique number that is not in the range of $f$.
4. Sum of values of a shifted odd function ●●●○○ 50 ptsGiven that $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(1) = 1$. If the graph of $f(x)$ is translated one unit to the right, the resulting graph is that of an even function. Find the sum $f(1) + f(2) + f(3) + \ldots + f(2009)$.
5. Function with midpoint averaging property ●●●○○ 50 ptsLet $f$ be a differentiable function satisfying $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ for all real $x$ and $y$. Given that $f(0)=1$ and $f'(0)=-1$, find $f(2)$.