Question

Care dintre functiile f:D->R sunt functii pare si care sunt impare :
a) f(x)=5x la puterea 4 -11,x€x
b) f(x)= radical din 25 -4x la puterea 2,x€(-1,2)
c) f(x)=x la puterea 3 +x/x la puterea 2 +4,x€R
d) f(x)=|x-4|-|4+x|,x€R
e) f(x)=-x-3x la puterea 3 /3+|x| , x€R

Answer 1

$f:D\rightarrow \mathbb_{R}, \\ \\ a) f(x) = 5x^4 - 11 \\ f(-x) = 5(-x)^4-11 = 5x^4-11 \Rightarrow f(-x) = f(x) \\ \\ \Rightarrow f para. \\ \\ b) f(x) = \sqrt{25}-4x^4 \\ f(-x) = 5 - 4(-x)^4 = 5 - 4x^4 \Rightarrow f(-x) = f(x) \\ \\ \Rightarrow f para.\\ \\ c) f(x) = \dfrac{x^3+x}{x^2+4}\\\\ f(-x) = \dfrac{(-x)^3+(-x)}{(-x)^2+4} = \dfrac{-x^3-x}{x^2+4} = -\dfrac{x^3+3}{x^2+4} \Rightarrow f(x) = -f(x) \\ \\ \Rightarrow f impara.$

$d) f(x) = |x-4|-|4+x| \\ f(-x) = |-x-4| - |4+(-x)| = |-(x+4)|-|4-x| = \\ =|x+4|-|-(x-4)| = |x+4| - |x-4| = |4+x| - |x-4| = \\ =-\Big(|x-4|-|4+x|\Big) \Rightarrow f(-x) = -f(x) \\ \\ \Rightarrow f impara.\\ \\ e) f(x) = \dfrac{-x-3x^3}{3+|x|} \\ f(-x) =\dfrac{-(-x) -3(-x)^3}{3+|-x|} = \dfrac{x-3\cdot (-1)\cdot x^3}{3+|x|} = \dfrac{x+3x^3}{3+|x|} = \\ =-\dfrac{-x-3x^3}{3+|x|}\Rightarrow f(-x) = -f(x) \\ \\ \Rightarrow f impara.$