Search problems

3 problems match “even functions”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. 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 $f(3)$.
2. Possible values of f(2018) ●●●○○ 50 ptsLet $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be an even function with $f(1)=1$ and $f(2017) \neq 1$. For all integers $x, y$, $2f(x+y)-f(x)-f(y) \leqslant |f(x)-f(y)|$. Find the value of $f(2018)$.
3. Unique solution with a parameter ●●●○○ 50 ptsFind all values of the parameter $a$ for which the equation $a x^{2}+\sin ^{2} x=a^{2}-a$ has a unique solution. Give the value of $a$.