Even functions

The values of even function at x=x and x=-x are same.

Definition 1: Even function

A function f(x) is said to be “even” if for every “x”, there exists “-x” in the domain of the function such that :

f - x = f x f - x = f x

An even function is symmetric about y-axis. If we consider the axis as a mirror, then the plot in first quadrant has its mirror image (bilaterally inverted) in second quadrant. Similarly, the plot in fourth quadrant has its mirror image (bilaterally inverted) in third quadrant.

Some examples of even functions are x 2 , | x | and cos x x 2 , | x | and cos x . In each case, we see that :

⇒ f - x = - x 2 = x 2 = f x ⇒ f - x = - x 2 = x 2 = f x

⇒ f - x = | - x | = | x | = f x ⇒ f - x = | - x | = | x | = f x

⇒ f - x = cos - x = cos x = f x ⇒ f - x = cos - x = cos x = f x

The right side is mirror image of left hand side and the left side is mirror image of right hand side of the curve.

Figure 1: Examples of even functions.

Even functions

It is important to see that if we rotate the curve by 180° about y-axis, then the appearance of the rotated curve is same as the original curve. We can state this alternatively as : if we rotate left hand side of the curve by 180° about y-axis, then we get the right hand curve and vice-versa.

Examples

Odd functions

The values of odd function at x=x and x=-x are equal in magnitude but opposite in sign.

Definition 2: Odd function

A function f(x) is said to be “odd” if for every “x”, there exists “-x” in the domain of the function such that :

f - x = - f x f - x = - f x

An odd function is symmetric about origin of the coordinate system. The plot in first quadrant has its mirror image (bilaterally inverted) in third quadrant. Similarly, the plot in second quadrant has its mirror image (bilaterally inverted) in fourth quadrant.

Some examples of odd functions are : x , x 3 and sin x x , x 3 and sin x . In each case, we see that :

⇒ f - x = - x = - f x ⇒ f - x = - x = - f x

⇒ f - x = - x 3 = - x 3 = - f x ⇒ f - x = - x 3 = - x 3 = - f x

⇒ f - x = sin - x = - sin x = - f x ⇒ f - x = sin - x = - sin x = - f x

The upper curve of these functions is exactly same as the lower curve across x-axis.

Figure 2: Examples of odd functions.

Odd functions

It is important to see that if we rotate the curve by 180° about origin, then the appearance of the rotated curve is same as the original curve. In other words, if we rotate right hand side of curve by 180° about origin, then we get left side of the curve. Further, it is interesting to note that we obtain left hand part of the plot of odd function in two steps : (i) drawing reflection (mirror image) of right hand plot about y-axis and (ii) drawing reflection (mirror image) of “reflection drawn in step 1” about x-axis.

Figure 3: Odd function as two successive mirror images

Odd function plot

Examples

Mathematical operations and nature of function

It is easy to find the nature of function resulting from mathematical operations, provided we know the nature of operand functions. As already discussed, we check for following possibilities :

Based on above algorithm, we can determine the nature of resulting function. For example, let us determine the nature of "fog" function when “f” is an even and “g” is an odd function. By definition,

f o g - x = f g - x f o g - x = f g - x

But, “g” is an odd function. Hence,

⇒ g - x = - g x ⇒ g - x = - g x

Combining two equations,

⇒ f o g - x = f - g x ⇒ f o g - x = f - g x

It is given that “f” is even function. Therefore, f(-x) = f(x). Hence,

⇒ f o g - x = = f - g x = f g x = f o g x ⇒ f o g - x = = f - g x = f g x = f o g x

Therefore, resulting “fog” function is even function.

The nature of resulting function subsequent to various mathematical operations is tabulated here for reference :

------------------------------------------------------------------------------------
f(x)      g(x)      f(x) ± g(x)      f(x) g(x)        f(x)/g(x), g(x)≠0    fog(x) 
------------------------------------------------------------------------------------
odd       odd          odd             even                 even            odd
odd       even         Neither         odd                  odd             even
even      even         even            even                 even            even
------------------------------------------------------------------------------------

We should emphasize here that we need not memorize this table. We can always carry out particular operation and determine whether a particular operation results in even, odd or neither of two function types. We shall work with a division operation here to illustrate the point. Let f(x) and g(x) be even and odd functions respectively. Let h(x) = f(x)/g(x). We now substitute “x” by “-x”,

⇒ h - x = f - x g - x ⇒ h - x = f - x g - x

But f(x) is an even function. Hence, f(-x) = f(x). Further as g(x) is an odd function, g(-x) = - g(x).

⇒ h - x = f x - g x = - h x ⇒ h - x = f x - g x = - h x

Thus, the division, here, results in an odd function.

There is an useful parallel here to remember the results of multiplication and division operations. If we consider even as "plus (+)" and odd as "minus (-)", then the resulting function is same as that resulting from multiplication or division of plus and minus numbers. Product of even (plus) and odd (minus) is minus(odd). Product of odd (minus) and odd (minus) is plus (even). Similarly, division of odd (minus) by even (plus) is minus (odd) and so on.

Square of an even or odd function

The square of even or odd function is always an even function.

Properties of derivatives

1: If f(x) is an even differentiable function on R, then f’(x) is an odd function. In other words, if f(x) is an even function, then its first derivative with respect to "x" is an odd function.

2: If f(x) is an odd differentiable function on R, then f’(x) is an even function. In other words, if f(x) is an odd function, then its first derivative with respect to "x" is an even function.

Composition of a function

Every real function can be considered to be composed from addition of an even and an odd function. This composition is unique for every real function. We follow an algorithm to prove this as :

Let f(x) be a real function for x ∈ ∈ R. Then,

f x = f x + f - x } - f - x } f x = f x + f - x } - f - x }

Rearranging,

f x = 1 2 { f x + f - x } + 1 2 { f x − f - x } = g x + h x f x = 1 2 { f x + f - x } + 1 2 { f x − f - x } = g x + h x

Now, we seek to determine the nature of functions “g(x)” and “h(x). For “g(x)”, we have :

⇒ g - x = 1 2 [ f − x + f { - − x } ] = 1 2 { f − x + f x } = g x ⇒ g - x = 1 2 [ f − x + f { - − x } ] = 1 2 { f − x + f x } = g x

Thus, “g(x)” is an even function.

Similarly,

⇒ h - x = 1 2 [ f − x − f { - − x } ] = 1 2 { f − x − f x } = − h x ⇒ h - x = 1 2 [ f − x − f { - − x } ] = 1 2 { f − x − f x } = − h x

Clearly, “h(x)” is an odd function. We, therefore, conclude that all real functions can be expressed as addition of even and odd functions.

Even and odd extensions of function

A function has three components – definition(rule), domain and range. What could be the meaning of extension of function? As a matter of fact, we can not extend these components. The concept of extending of function is actually not a general concept, but limited with respect to certain property of a function. Here, we shall consider few even and odd extensions. Idea is to complete a function defined in one half of its representation (x>=0) with other half such that resulting function is either even or odd function.

       |f(x);  0≤x≤a
g(x) = |
       | f(-x); -a≤x<0

       | f(x);  0≤x≤a
g(x) = |
       | -f(x); -a≤x<0

Exercises