Is function $f(x)=\lg(x+\sqrt{x^2+1})$ an odd or even function?
For any real numbers $x$ and $y$, the following holds $$[f(x+y)]^2 = [f(x)]^2 + [f(y)]^2$$
Find the exact form of $f(x)$.
Let $f(x)$ be a polynomial with respect to $x$ and $$f(x+1)+f(x-1)=2x^2-4x$$ Find $f(x)$.
Find the function $f(x)$ such that $f(0)=1$, $f(\frac{\pi}{2})=2$, and for any $x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)\cos y$$
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any $m, n\in\mathbb{N}$, $$f(m+n)=f(m)+f(n)+mn$$
Determine $f(n)$.
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any integer $n \ge 2$, $$f(n)=f(n-1) + 2^{n-1}$$
Determine $f(n)$.
Find the range of function $f(x)=3^{-|\log_2x|}-4|x-1|$.
Let $f(x)=x^3 -x^2 -13x+24$. Find three pairs of $(x,y)$ such that if $y=f(x)$, then $x=f(y)$.
The function $f$ satisfies $f(0)=0$, $f(1)=1$, and $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ for all $x,y\in\mathbb{R}$. Show that $f(x)=x$ for all rational numbers $x$.
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that
$$f(yf(x)-x)=f(x)f(y)+2x$$
for all $x,\ y\in{\mathbb{R}}$.
Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and $|x - y|$ $\leq$ $|f(x) - f(y)|$ $\leq$ $3|x - y|$ for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}$.
Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:
- it is an even function
- $f(2)=f(-2)=0$
- $f(x) > 0$ when $-2 < x < 2$, and
- the maximum of $f(x)$ is achieved at $x=\pm 1$.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that
$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$
Let $f(x)$ be a second degree function satisfying $f(-2)=0$ and $2x \lt f(x) \le\frac{x^2+4}{2}$. Find the value of $f(10)$.