Determine Algebraically if the functions are even, odd or neither.
(1) F(x) = cuberoot{x}...I said neither. Is this correct?
(2) F(x) = 2x/|x|...I said odd. Is this correct?
You are correct about (2) but (1) is odd.
Because:
$\displaystyle f(x) = x^{\frac{1}{3}} $
$\displaystyle -f(-x) = -1(-x)^{\frac{1}{3}}$
$\displaystyle = -1((-1)^{\frac{1}{3}}(x)^{\frac{1}{3}}) $
$\displaystyle = ((-1)^3(-1))^{\frac{1}{3}}x^{\frac{1}{3}} $
$\displaystyle =((-1)(-1))^{\frac{1}{3}}x^{\frac{1}{3}} $
$\displaystyle = 1^{\frac{1}{3}}x^{\frac{1}{3}}$
$\displaystyle =x^{\frac{1}{3}}$
By definition, a function is odd if $\displaystyle f(x) = -f(-x) $