Which Of The Following Is An Even Function?A. F ( X ) = ∣ X ∣ F(x)=|x| F ( X ) = ∣ X ∣ B. F ( X ) = X 3 − 1 F(x)=x^3-1 F ( X ) = X 3 − 1 C. F ( X ) = − 3 X F(x)=-3x F ( X ) = − 3 X D. F ( X ) = X 3 F(x)=\sqrt[3]{x} F ( X ) = 3 X ​

by ADMIN 231 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 we replace x with -x in the function, the output remains the same. Even functions have several important properties, including symmetry about the y-axis and the fact that their graphs are mirror images of each other about the y-axis. In this article, we will explore which of the given functions 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 we replace x with -x in the function, the output remains the same. For example, consider the function f(x) = x^2. If we replace x with -x, we get f(-x) = (-x)^2 = x^2, which is the same as f(x). Therefore, f(x) = x^2 is an even function.

Properties of Even Functions

Even functions have several important properties, including:

  • Symmetry about the y-axis: The graph of an even function is symmetric about the y-axis. This means that if we reflect the graph of the function about the y-axis, we get the same graph.
  • f(-x) = f(x): As mentioned earlier, an even function satisfies the condition f(-x) = f(x) for all x in its domain.
  • Graphs are mirror images: The graphs of even functions are mirror images of each other about the y-axis.

Examples of Even Functions

Some examples of even functions include:

  • f(x) = x^2: As mentioned earlier, f(x) = x^2 is an even function.
  • f(x) = |x|: The absolute value function f(x) = |x| is also an even function.
  • f(x) = cos(x): The cosine function f(x) = cos(x) is an even function.

Which of the Following is an Even Function?

Now that we have discussed what an even function is and some examples of even functions, let's consider the options given in the problem:

A. f(x) = |x| B. f(x) = x^3 - 1 C. f(x) = -3x D. f(x) = ∛x

To determine which of these functions is an even function, we need to check if they satisfy the condition f(-x) = f(x) for all x in their domain.

Option A: f(x) = |x|

The absolute value function f(x) = |x| is an even function because it satisfies the condition f(-x) = f(x) for all x in its domain. If we replace x with -x, we get f(-x) = |-x| = |x|, which is the same as f(x).

Option B: f(x) = x^3 - 1

The function f(x) = x^3 - 1 is not an even function because it does not satisfy the condition f(-x) = f(x) for all x in its domain. If we replace x with -x, we get f(-x) = (-x)^3 - 1 = -x^3 - 1, which is not the same as f(x).

Option C: f(x) = -3x

The function f(x) = -3x is not an even function because it does not satisfy the condition f(-x) = f(x) for all x in its domain. If we replace x with -x, we get f(-x) = -3(-x) = 3x, which is not the same as f(x).

Option D: f(x) = ∛x

The function f(x) = ∛x is not an even function because it does not satisfy the condition f(-x) = f(x) for all x in its domain. If we replace x with -x, we get f(-x) = ∛(-x), which is not the same as f(x).

Conclusion

In conclusion, the only even function among the given options is f(x) = |x|. This is because it satisfies the condition f(-x) = f(x) for all x in its domain, and it has the properties of an even function, including symmetry about the y-axis and the fact that its graph is a mirror image of itself about the y-axis.

References

Frequently Asked Questions About Even Functions

In this article, we will answer some of the most frequently asked questions about even functions.

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 we replace x with -x in the function, the output remains the same.

Q: What are some examples of even functions?

A: Some examples of even functions include:

  • f(x) = x^2: The function f(x) = x^2 is an even function because it satisfies the condition f(-x) = f(x) for all x in its domain.
  • f(x) = |x|: The absolute value function f(x) = |x| is also an even function.
  • f(x) = cos(x): The cosine function f(x) = cos(x) is an even function.

Q: What are the properties of even functions?

A: Even functions have several important properties, including:

  • Symmetry about the y-axis: The graph of an even function is symmetric about the y-axis. This means that if we reflect the graph of the function about the y-axis, we get the same graph.
  • f(-x) = f(x): An even function satisfies the condition f(-x) = f(x) for all x in its domain.
  • Graphs are mirror images: The graphs of even functions are mirror images of each other about the y-axis.

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

A: To determine if a function is even or odd, you can use the following steps:

  1. Replace x with -x in the function.
  2. Simplify the expression.
  3. If the expression is equal to the original function, then the function is even.
  4. If the expression is equal to the negative of the original function, then the function is 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.

Q: Can a function be both even and odd?

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

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 that are symmetric about the y-axis, such as a pendulum.
  • Engineering: Even functions are used to design systems that are symmetric about the y-axis, such as bridges.
  • Computer Science: Even functions are used in algorithms that require symmetry, such as image processing.

Conclusion

In conclusion, even functions are an important concept in mathematics that have several real-world applications. By understanding the properties and behavior of even functions, we can better analyze and solve problems in physics, engineering, and computer science.

References