|
|
Understand even and odd functions graphically and analytically.
Interactive Tutorial
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
|
|
