Even and Odd Functions

Discuss the concept of even and oddfunctions graphically and analytically.

Even Functions

The graph of even functions are symmetric with respect to the y axis.
To proove analytically that a given function f is even, we need to proove that:
f(x) = f(-x)

graph of even functions

Odd Functions

The graph of even functions are symmetric with respect to the origin (0,0). To proove analytically that a given function f is odd, we need to proove that:
f(- x) = - f(x)

graph of odd functions

Example 1

The formulas of the even functions f, g, h and i are given (see graphs above). Show analytically that each of these functions satisfies the property of an even function: f(-x) = f(x)
Solution to Example 1
The formulae of the four functions f, g, h and i are given with the graphs of these functions. (see top graph above).
1) f(x) = x2
f(-x) = (- x)2 = (-x)(-x) = x2
From the above, f(-x) = f(x), hence f(x) is an even function.

2) g(x) = - | x | + 2
g(-x) = - | - x | + 2
The absolute value function function | x | is an even function, therefore
| - x | = | x |
we simplify g(-x) to
g(-x) = - | x | + 2
g(-x) = g(x), hence g(x) is an even function.

3) h(x) = 10 e- 0.2 x2
h(- x) = 10 e - 0.2 (- x) 2 = 10 e - 0.2 x 2
Since h(-x) = h(x), h(x) is an even function.

4) i(x) = 8 cos(x)
i(- x) = 8 cos( - x)
The cosine function is an even function, therefore
cos(-x) = cos(x)
which gives
i(- x) = 8 cos( - x) = 8 cos(x)
i(-x) = i(x) and therefore i(x) ia an even function.

Example 2

The formulas of the odd functions j, k, l and m are given (see graphs above). Show analytically that each of these functions satisfies the property of an odd function: f(- x) = - f(x)
Solution to Example 2
The formulae of the four functions j, k, l and m are given with the graphs of these functions. (see top graph above).
1) j(x) = x3
j(-x) = (- x)3 = (-x)(-x)(-x) = - x3
From the above, j(-x) = - j(x), hence j(x) is an odd function.


2) k(x) = - x
k(-x) = - (-x) = x
From the above, k(-x) = - k(x), hence k(x) is an odd function.


3) l(x) = 6 sin(x)
l(-x) = 6 sin( - x)
The sine function is an odd function, therefore
sin(-x) = - sin(x)
which gives
l(-x) = 6 sin( - x) = - 6 sin(x)
l(-x) = - l(x) and therefore l(x) ia an odd function.


3) m(x) = ex - e-x
m(- x) = e(- x) - e-(-x)
Simplify
= e- x - ex
and rewrite as
= - ( - e- x + ex )
m(-x) = - m(x) and therefore m(x) is an odd function.

Exercises

Verify analytically whether each of these functions is even, odd or neither?
1. f(x) = - 2 x2 + 4
2. g(x) = 3 | x | - x2
3. h(x) = 1 / x
4. i(x) = tan(x)
5. j(x) = x2 + 3 x Detailed solutions to Above Exercises
1. f(x) = - 2 x2 + 4
Let us find f(-x) = - 2 (-x)2 + 4 = - 2 x2 + 4
f(-x) is equal to f(x) so function f is even.

2. g(x) = 3 | x | - x2
g(-x) = 3 | - x | - (-x)2 = 3 | x | - x2 and is equal to g(x) hence function g is even.

3. h(x) = 1 / x
h(-x) = 1 / (-x) = - 1 / x and is equal to - h(x), and therefore h is odd.

4. i(x) = tan(x)
i(-x) = tan(-x) = - tan(x), function i is odd.

5. j(x) = x2 + 3 x
j(-x) = (- x)2 + 3 (- x) = x2 - 3 x
j(-x) is not equal to j(x) and j(-x) is not equal to - j(x). Therefore j(x) is neither an even nor an odd function.

More References and Links

Properties of Trigonometric Functions
Applications, Graphs, Domain and Range of Functions

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