EVEN AND ODD FUNCTIONS
There are different types of functions in mathematics that we study. We can determine whether a function is even or odd algebraically or graphically. Even and Odd functions can be checked by plugging in the negative inputs (-x) in place of x into the function f(x) and considering the corresponding output value. Even and odd functions are classified on the basis of their symmetry relations. Even and odd functions are named based on the fact that the power function f(x) = xn is an even function, if n is even, and f(x) is an odd function if n is odd.
Let us explore other even and odd functions and understand their properties, graphs, and the use of even and odd functions in integration. A function can be even or odd or both even and odd, or neither even nor odd. Let's explore various examples to understand the concept.
What are Even and Odd Functions?
Generally, we consider a real-valued function to be even or odd. To identify if a function is even or odd, we plug in -x in place of x into the function f(x), that is, we check the output
value of f(-x) to determine the type of the function. Even and odd functions are symmetrical. Let us first understand their definitions.
Even and Odd Functions Definition
Even Function - For a real-valued function f(x), when the output value of f(-x) is the same as f(x), for all values of x in the domain of f, the function is said to be an even function.
An even function should hold the following equation: f(-x)
= f(x), for all values of x in D(f), where D(f) denotes the domain of the function f. In other words, we can say that the equation f(-x) - f(x) = 0 holds for an even function, for all x.
Let us consider an example, f(x) = x2.
f(-x) = (-x)2 = x2 for all values of x, as the square of a negative number is the same as the square of the positive value of the number. This implies f(-x) = f(x), for all x. Hence, f(x) = x2 is an even function. Similarly, functions like x4, x6, x8, etc.
are even functions.
Odd Function - For a real-valued function f(x), when the output value of f(-x) is the same as the negative of f(x), for all values of x in the domain of f, the function is said to be an odd function. An odd function should hold the following
equation: f(-x) = -f(x), for all values of x in D(f), where D(f) denotes the domain of the function f. In other words, we can say that the equation f(-x) + f(x) = 0 holds for an odd function, for all x
Let us consider an example, f(x) = x3.
f(-x) = (-x)3 = -(x3) for all values of x, as the cube of a negative number is the same as the negative of the cube of the positive value of the number. This implies f(-x) = -f(x), for all x. Hence, f(x) = x3 is an odd function. Similarly, functions like x5, x7, x9 etc. are odd functions.
Both Even and Odd Functions - A real-valued function f(x) is said to be both even and odd if it satisifies f(-x) = f(x) and f(-x) = -f(x) for all values of x in the domain of the function f(x). There is only one function which is both even and odd and that is the zero function, f(x) = 0 for all x. We know that for zero function, f(-x) = -f(x) = f(x) = 0, for all values of x. Hence, f(x) = 0 is an even and odd function.
Neither Even Nor Odd Function - A real-valued function f(x) is said to be neither even nor odd if it does not satisfy f(-x) = f(x) and f(-x) = -f(x) for atleast one value of x in the domain of the function f(x). Let us consider an
example to understand the definition better. Consider f(x) = 2x5 + 3x2 + 1, f(-x) = 2(-x)5 + 3(-x)2 + 1 = -2x5 + 3x2 + 1 which is neither equal to f(x) nor -f(x). Hence, f(x) = 2x5 + 3x2 + 1 is neither even nor odd function.
Even and Odd Functions in Trigonometry
In this section, we will segregate trigonometric functions as even and odd functions. We have six trigonometric ratios (sine, cosine, tangent, cotangent, cosecant, and secant).
These trigonometric ratios give positive values in different quadrants for various measures of angles.
In the first quadrant (where x and y coordinates are all positive), all six trigonometric ratios have positive values.
In the second quadrant, only sine and cosecant are positive.
In the third quadrant, only tangent and cotangent are positive. In the fourth quadrant, only cosine and secant are positive. Based on these signs, we will categorize them as even and odd functions.
If a trigonometric ratio is even or odd can be checked through a unit circle. An angle measured in anticlockwise direction is a positive angle whereas the angle measured in the clockwise direction is a negative angle.
• sinθ = y, sin(-θ) = -y; Therefore, sin(-θ) = -sinθ.
Hence, sinθ is an odd function.
• cosθ = y, cos(-θ) = y; Therefore, cos(-θ) = cosθ.
Hence, cosθ is an even function.
• tanθ = y, tan(-θ) = -y; Therefore, tan(-θ) = -tanθ.
Hence, tanθ is an odd function.
• cosecθ = y, cosec(-θ) = -y; Therefore, cosec(-θ) = - cosecθ. Hence, cosecθ is an odd function.
• secθ = y, sec(-θ) = y; Therefore, sec(-θ) = secθ.
Hence, secθ is an even function.
• cotθ = y, cot(-θ) = -y; Therefore, cot(-θ) = -cotθ.
Hence, cotθ is an odd function.
Integral Properties of Even and Odd Functions
The integral of a function gives the area below the curve. We use properties of even and odd functions while solving definite integrals. For that, we need to know the limits of the integral and the nature of the function. If the function is even or odd, and the interval is [-a, a], we can apply the following two rules:
When f(x) is even, ∫ 𝑓(𝑥)𝑑𝑥 = 2 ∫ 𝑓(𝑥)𝑑𝑥 −𝑎𝑎 0𝑎 When f(x) is odd, ∫ 𝑓(𝑥)𝑑𝑥 = 0−𝑎𝑎
Even and Odd Functions Graph
Let us now see how even and odd functions behave graphically. The graph of an even function is symmetric with respect to the y-axis. In other words, the graph of an even function remains the same after reflection about the y- axis. For any two opposite input values of x, the function value will remain the same all along the curve.
Whereas the graph of an odd function is symmetric with respect to the origin. In other words, the graph of an odd
function is at the same distance from the origin but in opposite directions. For any two opposite input values of x, the function has opposite y values. Here are a few examples of even and odd functions.
Figure 1. Even Function Graph
Figure 2. Odd Function Graph
Properties of Even and Odd Functions
• The sum of two even functions is even and the sum of two odd functions is odd.
• The difference between two even functions is even and the difference between two odd functions is odd.
• The sum of an even and odd function is neither even nor odd unless one of them is a zero function.
• The product of two even functions is even and the product of two odd functions is also an even function.
• The product of an even and an odd function is odd.
• The quotient of two even functions is even and the quotient of two odd functions is also an even function.
• The quotient of an even and an odd function is odd.
• The composition of two even functions is even and the composition of two odd functions is odd.
• The composition of an even and an odd function is even.
Important Notes on Even and Odd Functions
• A function f(x) is even if f(-x) = f(x), for all values of x in D(f) and it is odd if f(-x) = -f(x), for all values of x.
• In trigonometry, cos(θ) and sec(θ) are even functions, and sin(θ), cosec(θ), tan(θ), cot(θ) are odd functions.
• The graph even function is symmetric with respect to the y-axis and the graph of an odd function is symmetric about the origin.
• f(x) = 0 is the only function that is an even and odd function.
