Property 7
\(f:\mathbb R\to\mathbb R\)
\(f(-x)=-f(x)\)
function "f" is called odd
\(f(x)=x^3\)
\(f(x)=\cot x\)
\(f(x)=\tan x\)
\(f(x)=\sin x\)
\(f(x)=\operatorname{sgn}(x)\)
\(f(x)=k\cdot x^{2n-1}\qquad n\in\mathbb N\quad k\in\mathbb R\)
Odd functions are symmetrical according to the origin.
functions that are made of odd powers of "x" are odd functions.
\(f(x)=x\)
\(f(x)=x^3\)
\(f(x)=2x^5-3x^3+x\)
→ f(x) odd function
\(f(x):\mathbb R\to\mathbb R\)
1.
\(f(x)=(a-2)x^4+(b+2)x^2+abx\)
\(\Rightarrow f(x)=?\)
2.
\(f(x)+2f(-x)=x^3+2x\)
\(\Rightarrow f(2)=?\)
3.
\(f(x)=mx^3+(m+4)x^2+7x\)
\(\Rightarrow f(-1)=?\)
4.
\(f(x)=5f(-x)+x^3-3x\)
\(\Rightarrow f(-3)=?\)
5.
\(f(x)=\tan^3(x)+\sin^5(x)\)
\(\Rightarrow f(\samefrac\pi3)+f(-\samefrac\pi3)=?\)
6.
\(f(4)=2\)
\(g(x)=2f(-x)-x^3\)
\(\Rightarrow g(4)=?\)