Which Statement Best Describes How To Determine Whether $f(x)=9-4x^2$ Is An Odd Function?A. Determine Whether $9-4(-x)^2$ Is Equivalent To $9-4x^2$.B. Determine Whether $9-4(-x^2$\] Is Equivalent To

Introduction

In mathematics, an odd function is a function that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. This property is crucial in various mathematical and real-world applications, such as signal processing, image analysis, and data analysis. In this article, we will explore how to determine whether a given function is odd or not, using the function $f(x) = 9 - 4x^2$ as an example.

Understanding Odd Functions

An odd 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 resulting expression should be equal to the negative of the original function. In other words, if $f(x)$ is an odd function, then $f(-x) = -f(x)$.

Determining Whether a Function is Odd

To determine whether a function is odd, we need to check whether it satisfies the condition $f(-x) = -f(x)$. We can do this by substituting $-x$ into the function and simplifying the resulting expression. If the resulting expression is equal to the negative of the original function, then the function is odd.

Example: Determining Whether $f(x) = 9 - 4x^2$ is an Odd Function

Let's use the function $f(x) = 9 - 4x^2$ as an example. To determine whether this function is odd, we need to substitute $-x$ into the function and simplify the resulting expression.

Step 1: Substitute $-x$ into the function

We start by substituting $-x$ into the function $f(x) = 9 - 4x^2$. This gives us:

$f(-x) = 9 - 4(-x)^2$

Step 2: Simplify the resulting expression

Now, we need to simplify the resulting expression. We can do this by expanding the square and simplifying the resulting expression.

$f(-x) = 9 - 4(-x)^2$ $f(-x) = 9 - 4(-x)(-x)$ $f(-x) = 9 - 4x^2$

Step 3: Check whether the resulting expression is equal to the negative of the original function

Now, we need to check whether the resulting expression is equal to the negative of the original function. We can do this by comparing the resulting expression with the negative of the original function.

$f(-x) = 9 - 4x^2$ $-f(x) = -9 + 4x^2$

As we can see, the resulting expression $f(-x) = 9 - 4x^2$ is not equal to the negative of the original function $-f(x) = -9 + 4x^2$. Therefore, the function $f(x) = 9 - 4x^2$ is not an odd function.

Conclusion

In conclusion, to determine whether a function is odd, we need to check whether it satisfies the condition $f(-x) = -f(x)$. We can do this by substituting $-x$ into the function and simplifying the resulting expression. If the resulting expression is equal to the negative of the original function, then the function is odd. In this article, we used the function $f(x) = 9 - 4x^2$ as an example and showed that it is not an odd function.

Answer

Q: What is an odd function?

A: An odd 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 resulting expression should be equal to the negative of the original function.

Q: How do I determine whether a function is odd?

A: To determine whether a function is odd, you need to check whether it satisfies the condition $f(-x) = -f(x)$. You can do this by substituting $-x$ into the function and simplifying the resulting expression. If the resulting expression is equal to the negative of the original function, then the function is odd.

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

A: An odd function is a function that satisfies the condition $f(-x) = -f(x)$, while an even function is a function that satisfies the condition $f(-x) = f(x)$. In other words, if we replace $x$ with $-x$ in an odd function, the resulting expression is equal to the negative of the original function, while if we replace $x$ with $-x$ in an even function, the resulting expression is equal to the original function.

Q: Can a function be both odd and even?

A: No, a function cannot be both odd and even. If a function is odd, it satisfies the condition $f(-x) = -f(x)$, while if a function is even, 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 know whether a function is odd or even?

A: To determine whether a function is odd or even, you need to substitute $-x$ into the function and simplify the resulting expression. If the resulting expression is equal to the negative of the original function, then the function is odd. If the resulting expression is equal to the original function, then the function is even.

Q: Can I use a calculator to determine whether a function is odd or even?

A: Yes, you can use a calculator to determine whether a function is odd or even. Simply substitute $-x$ into the function and simplify the resulting expression using the calculator. If the resulting expression is equal to the negative of the original function, then the function is odd. If the resulting expression is equal to the original function, then the function is even.

Q: What are some examples of odd functions?

A: Some examples of odd functions include:

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

Q: What are some examples of even functions?

A: Some examples of even functions include:

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

Q: Can I use the properties of odd and even functions to simplify expressions?

A: Yes, you can use the properties of odd and even functions to simplify expressions. For example, if you have an expression of the form $f(x) + f(-x)$, you can simplify it using the fact that $f(-x) = -f(x)$ for an odd function, or $f(-x) = f(x)$ for an even function.

Conclusion

In conclusion, determining whether a function is odd or even is an important concept in mathematics. By understanding the properties of odd and even functions, you can simplify expressions and solve problems more efficiently. We hope this article has been helpful in answering your questions about odd and even functions.