Download Odd and Even Functions: Definition, Properties, and Graphs and more Lecture notes Pre-Calculus in PDF only on Docsity!
27 Odd and Even Functions
In this section we will focus on defining known functions as odd or even, and the implications of this, especially when sketching. By the end of this section, you should have the following skills:
- An understanding of the definition and properties of odd and even functions.
- Sketch graphs of odd and even functions.
- Evaluate definite integrals of odd and even functions.
27.1 Definition of Odd and Even Functions
We say a function f (x) is odd if f (−x) = −f (x) We say a function f (x) is even if f (−x) = f (x)
27.2 Odd Functions
- If f (x) is odd then f (0) = 0. This is true as f (−x) = −f (x) ⇒ f (0) = −f (0) ⇒ 2 f (0) = 0 ⇒ f (0) = 0.
- sin(x) is an odd function.
- x, x^3 , xn^ where n is an odd integer (negative or positive) are all odd.
- f (x) − f (−x) for any function f (x) is odd. Hence ex^ − e−x^ is odd.
- tan(x) is odd as tan(−x) = sin(−x)/cos(−x) = − sin(x)/ cos(x) = − tan(x)
- a multiple of an odd function is odd.
- sum of odd functions is odd.
- product or division of an odd and an even function is odd. For if f (x) is odd and g(x) is even then if h(x) = f (x)g(x) we have
h(−x) = f (−x)g(−x) = −f (x)g(x) = −h(x)
Similarly for division of odd by an even function or vice versa. Examples include functions such as x^2 sin(x), x^3 cos(x)
- f (g(x)) is odd when both f (x), g(x) are odd as
f (g(−x)) = f (−g(x)) = −f (g(x))
So sin(x^3 − x) is odd.
- The derivative of an even function is odd.
- The inverse of an odd function is odd (e.g. arctan(x) is odd as tan(x) is odd).
27.3 Even Functions
The following functions are even:
- cos(x)
- c a constant function, x^2 , xn^ where n is an even integer (negative or positive)
- f (x) + f (−x) for any function f (x). Hence ex^ + e−x^ is even.
- multiple of an even function
- sum of even functions. Hence x^4 − 2 x^2 + 5 is even.
- product or division of an odd and an odd function for if f (x) is odd and g(x) is odd then if h(x) = f (x)g(x) we have
h(−x) = f (−x)g(−x) = −(−f (x)g(x)) = h(x)
Also the product or division of two even functions is even. Examples of such even functions include the square f (x)^2 of any func- tion, x sin(x), x^2 cos(x) etc.
Sketch the graph for − 2 ≤ x ≤ 2 Solution
As f (x) is an odd function (it is an odd function of an odd func- tion) we first reflect in the x-axis to get
Reflect in x-axis
Then we reflect in the y-axis to obtain:
Reflect in y-axis
So putting the graphs together, the graph for − 2 ≤ x ≤ 2 is
Graph of sin(x^3 )
- Graph of even functions If you know the graph of the even function f (x) for positive values of x then it is a simple matter to get the graph for negative values by reflecting the graph in the y-axis. This follows as if (x, f (x)) lies on the graph of f (x) for positive values then on reflecting in the y-axis we have:
(x, f (x)) → (−x, f (x)) = (−x, f (−x))
and (−x, f (−x) is on the graph for negative values. Note we used the fact that f (−x) = f (x) here.
Example 2 Let f (x) = x^4 − 3 x^2 + 1. The graph of f (x) for 0 ≤ x ≤ 2 is
graph of x^4 − 3 x^2 + 1 for positive x
Sketch the graph for − 2 ≤ x ≤ 2 Solution
Proof Make the substitution u = −x ⇒ −du = dx in
I =
−a
f (x)dx
to get
I = −
a
f (−u)du
∫ (^) a
0
f (u)du
as f (−u) = f (u) Hence ∫ (^) a
−a
f (x)dx =
−a
f (x)dx +
∫ (^) a
0
f (x)dx
∫ (^) a
0
f (x)dx
27.5.2 Odd Functions
Let a ≥ 0. If f (x) is odd we have ∫ (^0)
−a
f (x)dx = −
∫ (^) a
0
f (x)dx ⇒ ∫ (^) a
−a
f (x)dx = 0
Proof Make the substitution u = −x ⇒ −du = dx in
I =
−a
f (x)dx
to get
I = −
a
f (−u)du
∫ (^) a
0
f (u)du
as f (−u) = −f (u) Hence ∫ (^) a
−a
f (x)dx =
−a
f (x)dx +
∫ (^) a
0
f (x)dx
= 0
27.5.3 Examples
Example 3 Evaluate the following definite integrals
(a)
I =
− 1
x^4 sin(x)dx
(b)
I =
− 2
(x^2 sin(3x) + x^5 + 3x^2 )dx
(c)
I =
− 1
arctan(x^3 + x) cos(x) x^4 + x^2 + 1
dx
Solution
(a) x^4 sin(x) is odd as it is the product of an odd and an even function. Hence I = 0
(b) x^2 sin(3x)+x^5 is an odd function as x^2 sin(3x) is odd as is x^5. Hence
I =
− 2
3 x^2 dx = 2
0
3 x^2 dx = 2[x^3 ]^20 = 16
as 3 x^2 is an even function.
(c) arctan(x^3 + x) cos(x) x^4 + x^2 + 1
(c) arcsin(tan(x)) sin(x) is an even function, x^3 + x is an odd function, hence arcsin(tan(x)) sin(x) x^3 + x is an odd function. It follows that
I =
− 2
arcsin(tan(x)) sin(x) x^3 + x
− 2
3 x^2 + 5 dx
= [x^3 + 5x]^2 − 2 = 36
