What is an even function? What is an odd function?

What is an even function ? Not only even functions , odd functions are also of great interest. Let's learn about these two concepts together!

Functions in mathematics can be classified into even and odd functions based on their symmetry along the y-axis. An even function is a function that remains constant when its input is negated (the output is the same for x and -x), reflecting symmetry around the y-axis. On the other hand, an odd function becomes negative when its input is negated, demonstrating symmetry around the origin. A function f is an even function if f(-x) = f(x), for all x in the domain of f. A function f is an odd function if f(-x) = -f(x) for all x in the domain of f, that is:

• Even function:f(-x) = f(x)
• Odd function:f(-x) = -f(x)

In this article, we will discuss in detail about even and odd functions, definition of even and odd functions, even and odd functions in trigonometry, graph of even and odd functions and many other contents and information you need to know.

Table of Contents

• What is an even function?
• What is an odd function?
• Graph of even and odd functions
• What is a function that is neither even nor odd?
• Remember a common odd-even function
• How to determine even and odd functions
• Exercises on examining the parity of functions

What is an even function?

The function y = f (x) with domain D is called an even function if it satisfies the following two conditions:

• ∀ x ∈ D ⇒ − x ∈ D
• ∀ x ∈ D : f ( − x ) = f ( x )

For example: The function y = x² is an even function.

What is an odd function?

The function y = f ( x ) with domain D is called an odd function if it satisfies the following two conditions:

• ∀ x ∈ D ⇒ − x ∈ D
• ∀ x ∈ D : f (−x)= − f(x)

Example: Example: The function y = x is an odd function.

Note. The first condition is called the domain-symmetric condition about 0.

For example, D = (-2;2) is a set symmetric about 0, while the set D' = [-2;3] is not symmetric about 0.

The set R = (−∞;+∞) is a symmetric set.

Note: A function does not have to be even or odd.

For example: The function y = 2x + 1 is neither an even function nor an odd function because:

At x = 1 we have f(1) = 2.1 + 1 = 3

At x = -1 we have f(-1) = 2.(-1) + 1 = -1

→ The two values ​​f(1) and f(-1) are neither equal nor opposite.

Graph of even and odd functions

Even functions have graphs that take the y-axis as the axis of symmetry.

The odd function has a graph with the origin O as the center of symmetry.

What is a function that is neither even nor odd?

Not every function can be defined as even or odd. Some functions are neither even nor odd, such as: y=x²+x, y=tan(x-1),…

In addition, there is a special type of function that is both even and odd. For example, the function y=0

Remember a common odd-even function

Even function

y = ax2 + bx + c if and only if b = 0

Quadratic function

y = cosx

y = f(x)

Odd function

y = ax + b if and only if b = 0

y = ax3 + bx2 + cx + d if and only if b = d = 0

y = sinx; y = tanx; y = cotx

Some other cases

F(x) is an even function and has a derivative on its domain, then its derivative is an odd function.

F(x) is an odd function and has a derivative on its domain, then its derivative is an even function.

An odd degree polynomial function is not an even function.

Even degree polynomial functions are not odd functions.

How to determine even and odd functions

To determine the odd-even function, we perform the following steps:

Step 1: Find the domain: D

If ∀x ∈ D ⇒ -x ∈ D Go to step three

If ∃ x0 ∈ D ⇒ -x0 ∉ D, then the function is neither even nor odd.

Step 2: Replace x with -x and calculate f(-x)

Step 3: Examine the sign (compare f(x) and f(-x)):

° If f(-x) = f(x) then the function f is even

° If f(-x) = -f(x) then the function f is odd

° Other cases: function f does not have parity

Exercises on examining the parity of functions

Lesson 4 page 39 Algebra 10 Textbook: Consider the odd-even properties of the following functions:

a) y = |x|;

b) y = (x + 2)2;

c) y = x3 + x;

d) y = x2 + x + 1.

Prize

a) Let y = f(x) = |x|.

° TXĐ: D = R so for ∀x ∈ D then –x ∈ D.

° f(–x) = |–x| = |x| = f(x).

→ So the function y = |x| is an even function.

b) Let y = f(x) = (x + 2)2.

° TXĐ: D = R so for ∀x ∈ D then –x ∈ D.

° f(–x) = (–x + 2)2 = (x – 2)2 ≠ (x + 2)2 = f(x)

° f(–x) = (–x + 2)2 = (x – 2)2 ≠ – (x + 2)2 = –f(x).

→ So the function y = (x + 2)2 is neither even nor odd.

c) Let y = f(x) = x3 + x.

° TXĐ: D = R so for ∀x ∈ D then –x ∈ D.

° f(–x) = (–x)3 + (–x) = –x3 – x = – (x3 + x) = –f(x)

→ So y = x3 + x is an odd function.

d) Let y = f(x) = x2 + x + 1.

° TXĐ: D = R so for ∀x ∈ D then –x ∈ D.

° f(–x) = (–x)2 + (–x) + 1 = x2 – x + 1 ≠ x2 + x + 1 = f(x)

° f(–x) = (–x)2 + (–x) + 1 = x2 – x + 1 ≠ –(x2 + x + 1) = –f(x)

→ So the function y = x2 + x + 1 is neither even nor odd.

Is there a function defined on R that is both an even and an odd function?...

Prize:

It is easy to see that the function y = 0 is a function defined on R, both an even and an odd function.

Suppose the function y = f (x) is any function with such properties. Then for every x in R we have:

F (–x) = f (x) (because f is an even function);

F (–x) = – f (x) (because f is an odd function).

From this we can deduce that for every x in R, f(x)=−f(x), meaning f(x)=0. So y=0 is the only function defined on R that is both an even and an odd function.

Frequently asked questions about even and odd functions

What are even and odd functions?

If f(x) = f(−x) for all x in their domain, then even functions are symmetric about the y-axis. Odd functions are symmetric about the origin, meaning that for all x in their domain, f(−x) = −f(x).

How to know if a function is even or odd?

A function is even if f(-x) = f(x), and odd if f(-x) = -f(x) for every element in the domain of f. If it does not satisfy any of these properties, then it is neither odd nor even.

What is the difference between odd and even periodic functions?

Difference between odd and even periodic functions: An even function satisfies f(−x) = f(x) for all x in the domain, while an odd function satisfies f(−x) = −f(x).

In addition to even and odd functions, you can learn some other important mathematical knowledge such as square numbers , irrational numbers, rational numbers , prime numbers , natural numbers ... in the Education section of Quantrimang.com.