Graph is symmetric with respect to the origin

Oct 13, 2025 by wbrum.org 19 views

Introduction

In mathematics, an even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if a function is even, then its graph will be symmetric with respect to the y-axis. In this article, we will examine four different functions and determine which one is an even function.

What is an Even Function?

An even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if a function is even, then its graph will be symmetric with respect to the y-axis. For example, the function f(x) = x^2 is an even function because f(-x) = (-x)^2 = x^2.

Function 1: f(x) = |x|

The first function we will examine is f(x) = |x|. This function represents the absolute value of x, which is the distance of x from 0 on the number line. To determine if this function is even, we need to evaluate f(-x).

f(-x) = |-x| = |x| = f(x)

Since f(-x) = f(x), we can conclude that f(x) = |x| is an even function.

Function 2: f(x) = x^3 - 1

The second function we will examine is f(x) = x^3 - 1. To determine if this function is even, we need to evaluate f(-x).

f(-x) = (-x)^3 - 1 = -x^3 - 1 ≠ x^3 - 1 = f(x)

Since f(-x) ≠ f(x), we can conclude that f(x) = x^3 - 1 is not an even function.

Function 3: f(x) = -3x

The third function we will examine is f(x) = -3x. To determine if this function is even, we need to evaluate f(-x).

f(-x) = -3(-x) = 3x ≠ -3x = f(x)

Since f(-x) ≠ f(x), we can conclude that f(x) = -3x is not an even function.

Function 4: f(x) = ∛x

The fourth function we will examine is f(x) = ∛x. To determine if this function is even, we need to evaluate f(-x).

f(-x) = ∛(-x) ≠ ∛x = f(x)

Since f(-x) ≠ f(x), we can conclude that f(x) = ∛x is not an even function.

Conclusion

In conclusion, the only even function among the four functions examined is f(x) = |x|. This function satisfies the condition f(-x) = f(x) for all x in its domain, and its graph is symmetric with respect to the y-axis.

Why is f(x) = |x| an Even Function?

f(x) = |x| is an even function because it satisfies the condition f(-x) = f(x) for all x in its domain. This means that if we replace x with -x in the function, we get the same result. In other words, f(-x) = |-x| = |x| = f(x). This symmetry with respect to the y-axis is a characteristic of even functions.

What are the Properties of Even Functions?

Even functions have several properties that make them useful in mathematics and other fields. Some of the properties of even functions include:

Symmetry with respect to the y-axis: Even functions are symmetric with respect to the y-axis, which means that their graph will be the same if we reflect it across the y-axis.
f(-x) = f(x): Even functions satisfy the condition f(-x) = f(x) for all x in their domain.
f(x) = f(-x): Even functions also satisfy the condition f(x) = f(-x) for all x in their domain.
Graph is symmetric with respect to the origin: Even functions have a graph that is symmetric with respect to the origin, which means that their graph will be the same if we reflect it across the origin.

Examples of Even Functions

Some examples of even functions include:

f(x) = x^2: This function is an even function because f(-x) = (-x)^2 = x^2.
f(x) = |x|: This function is an even function because f(-x) = |-x| = |x|.
f(x) = cos(x): This function is an even function because f(-x) = cos(-x) = cos(x).

Conclusion

Q: What is an even function?

A: An even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if a function is even, then its graph will be symmetric with respect to the y-axis.

Q: Why is f(x) = |x| an even function?

A: f(x) = |x| is an even function because it satisfies the condition f(-x) = f(x) for all x in its domain. This means that if we replace x with -x in the function, we get the same result. In other words, f(-x) = |-x| = |x| = f(x). This symmetry with respect to the y-axis is a characteristic of even functions.

Q: What are the properties of even functions?

A: Even functions have several properties that make them useful in mathematics and other fields. Some of the properties of even functions include:

Symmetry with respect to the y-axis: Even functions are symmetric with respect to the y-axis, which means that their graph will be the same if we reflect it across the y-axis.
f(-x) = f(x): Even functions satisfy the condition f(-x) = f(x) for all x in their domain.
f(x) = f(-x): Even functions also satisfy the condition f(x) = f(-x) for all x in their domain.
Graph is symmetric with respect to the origin: Even functions have a graph that is symmetric with respect to the origin, which means that their graph will be the same if we reflect it across the origin.

Q: What are some examples of even functions?

A: Some examples of even functions include:

f(x) = x^2: This function is an even function because f(-x) = (-x)^2 = x^2.
f(x) = |x|: This function is an even function because f(-x) = |-x| = |x|.
f(x) = cos(x): This function is an even function because f(-x) = cos(-x) = cos(x).

Q: How do I determine if a function is even?

A: To determine if a function is even, you need to evaluate f(-x) and compare it to f(x). If f(-x) = f(x), then the function is even. If f(-x) ≠ f(x), then the function is not even.

Q: What are some real-world applications of even functions?

A: Even functions have several real-world applications, including:

Physics: Even functions are used to describe the motion of objects in physics, such as the motion of a pendulum or a spring.
Engineering: Even functions are used in engineering to describe the behavior of systems, such as the behavior of a circuit or a mechanical system.
Computer Science: Even functions are used in computer science to describe the behavior of algorithms, such as the behavior of a sorting algorithm.

Q: Can a function be both even and odd?

A: No, a function cannot be both even and odd. If a function is even, then it satisfies the condition f(-x) = f(x) for all x in its domain. If a function is odd, then it satisfies the condition f(-x) = -f(x) for all x in its domain. These two conditions are mutually exclusive, so a function cannot be both even and odd.

Q: What is the difference between an even function and an odd function?

A: The main difference between an even function and an odd function is the way they behave when we replace x with -x. An even function satisfies the condition f(-x) = f(x) for all x in its domain, while an odd function satisfies the condition f(-x) = -f(x) for all x in its domain. This difference in behavior leads to different properties and applications for even and odd functions.