# Even and Odd Functions

 Understand even and oddfunctions graphically and analytically. Interactive Tutorial Your browser is completely ignoring the tag! click on the button above "click here to start" and MAXIMIZE the window obtained. Check all 8 functions f, g, h, i, j, k, l and m in the panel of the applet one by one and examine their graphs. Which graphs are symmetric with respect to y axis? Which graphs are symmetric with respect to the origin of the system of axes (0 , 0)? The graphs that are symmetric with respect to y axis correspond to even functions and the graphs that are symmetric with respect to origin correspond to odd functions. Classify each of the 8 functions as even or odd. The formulas of functions f, g, h, i, j, k, l and m used in the applet are given below: f(x) = x 2 g(x) = x 3 h(x) = x i(x) = - | x | + 2 j(x) = x 2 + 2 k(x) = cos x l(x) = sin x m(x) = - x 3 Verify analytically that all even functions satisfy the condition f(x) = f(-x) and all odd functions satisfy the condition f(x) = - f(-x) Exercises: Verify whether each of these functions is even, odd or neither? 1. f(x) = 2 x 2. g(x) = | x | + 2 3. h(x) = 1 / x 4. i(x) = x 2 Detailed solutions to Above Exercises: 1. f(x) = 2 x Let us find f(-x) = 2(-x) = -2 x f(-x) is not equal to f(x) so function f is not even. However -f(-x) = 2x and is equal to f(x) and therefore function f is odd. 2. g(x) = | x | + 2 g(-x) = | -x | + 2 = | x | + 2 and is equal to g(x) hence function g is even. 3. h(x) = 1 / x h(-x) = 1 / (-x) = - 1 / x and is not equal to h(x). However -h(-x) = 1 / x = h(x) and therefore h is odd. 4. i(x) = x 2 i(-x) = (-x) 2 = x 2 , function i is even. More tutorials on functions. Applications, Graphs, Domain and Range of Functions