0. Functions

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

State whether the functions represented by graphs A , B , C and in the figure are even, odd, or neither. <IMAGE>

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x^4+5x^2-12$

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x{}^5-x^3-2$

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x\mid x\mid$

Identify the symmetry (if any) in the graphs of the following equations.

$y^2-4x^2=4$

Given the graph of the following function, determine the intervals on which $f\left(x\right)$  is increasing.