Even Functions

Introduction

Functions are the basics to understand calculus mathematics. They are essential fundamental for applied mathematics in our daily life. There are two types of functions, i.e. even and odd functions. Both functions satisfy specific symmetry relations that may be concerning their additive inverse. These functions are an essential part of the mathematical analysis of power series and Fourier series theory. We can identify even functions algebraically and graphically. As the name suggests, the power function f(x) = xn, n is always even in even functions.

On the other hand, for abnormal function, n is always odd. An example for function (even) shall be the trigonometric even function. Let’s explore these even functions in study material in detail.

Even Functions

If f is a real-valued function for a real set,

f shall be even only if:

F(x) = f (-x)

It can also state as, f(x)-f (-x) =0

So, the functions that obey the above rule are the even functions in mathematics.

The graph for an even function will be symmetric with respect to its y-axis. Therefore, the graph does not change after reflecting on the y-axis.

Application of Even Function in Trigonometry

As we know, there are a total 6 trigonometric ratios, i.e. sin, cosine, tangent, cotangent, cosecant, and secant. Based on their sign, i.e. either positive or negative, we categorise them into even and odd functions. The angles measured in an anticlockwise direction will be positive angles.

• Conditions of Even Functions in Trigonometry:

a) Cosθ=x, cos(-θ)=x. Thus, cos(-θ)= cos(θ)

This shows that cosθ is an even function.

b) secθ = y, sec(-θ) = y. Thus, sec(-θ) = secθ.

This shows that secθ is also an even function.

Properties of Even Functions

The characteristic features of even functions are as follows:

• In the addition of two even functions, we always get even.
• In the same way, on subtracting two even functions, we will get even too.
• The sum of an even and odd will be neither even nor odd unless one is equal to zero domain.
• Like addition or subtraction, the product of two even functions will also be an even.
• However, the product of an even and odd will be only odd.
• The quotient of two even functions on the division will also be even.
• Like the product property, the quotient of an even and odd function will also be odd.
• The composition for two even functions will also be an even.
• Unlike the product and division property, the composition of one even and odd function will be only even.
• Other than the above properties, the derivative of an even will always be odd.
• The integral of an even function, say –B to B, will be twice the integral from 0 to +B.
• The Maclaurin series of any even function will only have even powers.
• Similarly, the Fourier series of periodic even functions will only have cosine trigonometric terms.

Questions on Even Functions

Q 1: Find whether the function f(x) = cosx is even or odd?

Solution: According to question, the given function f(x) = cosx

= (ex + e-x)/2

For determining, whether f(x) is even or odd, we substitute –x instead of x in the given function.

f(-x) = cos(-x)

= (e-x + e-(-x))/2

= (e-x + ex)/2 = cosx = f(x).

Answer: So we can conclude that the given function, f(x) = cosx is an even function.

Q 2: A given function is as f(x) = x2. What will be the value of f(-x). Is the given function odd or even?

Solution: As we have already discussed above,

f(−x) = (−x)2 = x2 = f(x)

Thus we can simply conclude that, f(x) = x2 is an even function.

Verification: For the verification, take any value for x.

For x = 4, the value of f(x) will be as follows:

f(4) = 42 = 16

Now we will determine the value of f(−x) as:

f(−4) = (−4)2 = 16 = f(4)

Hence Verified.

Conclusion

There are two basic classification functions in mathematics, i.e. odd and even functions. Any function is stated as even only if f(x) = f (-x) for all x. In other words, we can say that even function shall be the same for the positive x-axis, negative y-axis, or may be symmetric about the y-axis. We can identify even functions algebraically and graphically. As the name suggests, the power function f(x) = xn, n is always even in even functions.

On the other hand, for odd functions n is always odd. The sum, difference and product of two even functions are always even. However, in the case of one even and one odd, the result may vary as per the condition. So, explore these study material notes on even functions.