Determine The Nature Of The Function F ( X ) = X 6 + 10 X 4 − 11 X 2 F(x)=x^6+10x^4-11x^2 F ( X ) = X 6 + 10 X 4 − 11 X 2 .A. Function F F F Is Neither Even Nor Odd. B. Function F F F Is Both Even And Odd. C. Function F F F Is Even. D. Function F F F Is Odd.

Introduction

In mathematics, functions are classified into two main categories: even and odd functions. Understanding the nature of a function is crucial in various mathematical applications, including calculus, algebra, and trigonometry. In this article, we will explore the concept of even and odd functions, and determine the nature of the given function $f(x)=x^6+10x^4-11x^2$.

What are Even and Odd Functions?

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 $-x$, the function remains unchanged. On the other hand, an odd function satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. This means that if we replace $x$ with $-x$, the function changes sign.

Properties of Even and Odd Functions

Even functions have the following properties:

• The graph of an even function is symmetric with respect to the y-axis.
• The function is unchanged when $x$ is replaced with $-x$.
• The function has even powers of $x$ in its expression.

Odd functions have the following properties:

• The graph of an odd function is symmetric with respect to the origin.
• The function changes sign when $x$ is replaced with $-x$.
• The function has odd powers of $x$ in its expression.

Determining the Nature of the Function $f(x)=x^6+10x^4-11x^2$

To determine the nature of the function $f(x)=x^6+10x^4-11x^2$, we need to check if it satisfies the conditions of an even or odd function.

Checking for Even Function

To check if the function is even, we need to replace $x$ with $-x$ and see if the function remains unchanged.

$f(-x) = (-x)^6 + 10(-x)^4 - 11(-x)^2$ $f(-x) = x^6 + 10x^4 - 11x^2$

Since $f(-x) = f(x)$, the function is even.

Checking for Odd Function

To check if the function is odd, we need to replace $x$ with $-x$ and see if the function changes sign.

$f(-x) = (-x)^6 + 10(-x)^4 - 11(-x)^2$ $f(-x) = x^6 + 10x^4 - 11x^2$

Since $f(-x) = f(x)$, the function is not odd.

Conclusion

Based on the analysis, we can conclude that the function $f(x)=x^6+10x^4-11x^2$ is an even function. The function satisfies the condition $f(-x) = f(x)$, which is the defining property of an even function.

Answer

The correct answer is:

C. Function $f$ is even.

Final Thoughts

Introduction

In our previous article, we explored the concept of even and odd functions, and determined the nature of the function $f(x)=x^6+10x^4-11x^2$. In this article, we will answer some frequently asked questions related to determining the nature of a function.

Q: What is the difference between an even and an odd 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 $-x$, the function remains unchanged. On the other hand, an odd function satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. This means that if we replace $x$ with $-x$, the function changes sign.

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

A: To determine if a function is even or odd, you need to replace $x$ with $-x$ and see if the function remains unchanged or changes sign. If the function remains unchanged, it is an even function. If the function changes sign, it is an odd function.

Q: What are some examples of even and odd functions?

A: Some examples of even functions include:

• $f(x) = x^2$
• $f(x) = \cos x$
• $f(x) = e^{x^2}$

Some examples of odd functions include:

• $f(x) = x^3$
• $f(x) = \sin x$
• $f(x) = \tan x$

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 $f(-x) = f(x)$, and if it is odd, it satisfies the condition $f(-x) = -f(x)$. These two conditions are mutually exclusive, so a function cannot satisfy both of them.

Q: How do I graph an even or odd function?

A: To graph an even function, you can use the following steps:

1. Graph the function $f(x)$.
2. Reflect the graph of $f(x)$ across the y-axis to get the graph of $f(-x)$.
3. The graph of $f(-x)$ is the same as the graph of $f(x)$.

To graph an odd function, you can use the following steps:

1. Graph the function $f(x)$.
2. Reflect the graph of $f(x)$ across the origin to get the graph of $f(-x)$.
3. The graph of $f(-x)$ is the negative of the graph of $f(x)$.

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

A: Even and odd functions have many real-world applications, including:

• Signal processing: Even and odd functions are used to analyze and process signals in various fields, including audio and image processing.
• Physics: Even and odd functions are used to describe the behavior of physical systems, including the motion of particles and the behavior of waves.
• Engineering: Even and odd functions are used to design and analyze systems, including electrical circuits and mechanical systems.

Conclusion

In conclusion, determining the nature of a function is an important concept in mathematics. By understanding the properties of even and odd functions, we can analyze and graph functions, and apply them to real-world problems. We hope that this Q&A article has provided you with a better understanding of even and odd functions.