Functions › Even, odd and periodic functions

Odd and even functions

Solve the first problem 18 problems

Value of an odd function ●○○○○ 10 ptsThe function $y=f(x)$ is an odd function, and $f(x)=3^x - 1$ for $x \geq 0$ . Find $f(-2)$ .
Odd function value at a positive point ●○○○○ 10 ptsGiven that $f(x)$ is an odd function, and when $x < 0$ , $f(x) = x^3 + x^2$ , find $f(2)$ .
Value of an odd function shifted by a constant ●○○○○ 10 ptsGiven that $f(x)$ is an odd function, $g(x)=f(x)+6$ , and $g(-1)=3$ . Find $f(1)$ .
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 $f(2)$ .
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 $f(3)$ .
Even function on an interval ●●○○○ 20 ptsLet $f(x)=x^{2}+3ax+4$ be defined on the interval $b-3 \leq x \leq 2b$ . Given that $f$ is an even function on this interval, find the value of $a-b$ .
Roots of a symmetric odd function ●●●○○ 50 ptsLet $y=f(x)$ be an odd function defined on $\mathbf{R}$ whose graph is symmetric about the line $x=1$ . When $0<x\leq 1$ , $f(x)=\log_3 x$ . Find the sum of all real roots of the equation $f(x)=-\frac{1}{3}+f(0)$ in the interval $(0,10)$ .
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 $f(1) + f(2) + f(3) + \ldots + f(2009)$ .
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 $f(2)$ .

More in Even, odd and periodic functions

Periodic functions Statements about functions Zeros of symmetric functions