Even and Odd Functions

Understand even and oddfunctions graphically and analytically.

Interactive Tutorial

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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

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