Property 7
Odd Function
\(f:\mathbb R\to\mathbb R,\quad f(-x)=-f(x)\)
Such a function is called odd. Examples include \(x^3\), \(\sin x\), \(\tan x\), \(\cot x\), \(\operatorname{sgn}(x)\), and polynomials made only from odd powers of \(x\). Odd functions are symmetric about the origin.
1.
\(f(x)=(a-2)x^4+(b+2)x^2+abx\)
\(f\text{ is odd}\)
\(f(x)=?\)
2.
\(f(x)+2f(-x)=x^3+2x\)
\(f(2)=?\)
3.
\(f(x)=mx^3+(m+4)x^2+7x\)
\(f\text{ is odd}\)
\(f(-1)=?\)
4.
\(f(x)=5f(-x)+x^3-3x\)
\(f(-3)=?\)
5.
\(f(x)=\tan^3x+\sin^5x\)
\(f(\samefrac\pi3)+f(-\samefrac\pi3)=?\)
6.
\(f(4)=2\)
\(g(x)=2f(-x)-x^3\)
\(g(4)=?\)