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

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Introduction

In mathematics, a function is classified as odd or even based on its behavior when the input is negated. An odd function satisfies the condition f(-x) = -f(x), while an even function satisfies the condition f(-x) = f(x). In this article, we will explore how to determine whether a given function is odd or even, using the function f(x) = 9 - 4x^2 as an example.

Understanding Odd and Even Functions

Before we dive into the example, let's briefly review the definitions of odd and even functions.

  • Odd Function: A function f(x) is said to be odd if f(-x) = -f(x) for all x in the domain of the function.
  • Even Function: A function f(x) is said to be even if f(-x) = f(x) for all x in the domain of the function.

Determining Whether a Function is Odd

To determine whether a function is odd, we need to check if f(-x) = -f(x). Let's apply this condition to the given function f(x) = 9 - 4x^2.

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 Expression

Now, let's simplify the expression f(-x) = 9 - 4(-x)^2. Using the property of exponents that (-a)^n = a^n for even n, we can rewrite (-x)^2 as x^2. Therefore, f(-x) = 9 - 4x^2.

Step 3: Compare with the Original Function

Now that we have simplified the expression f(-x) = 9 - 4x^2, let's compare it with the original function f(x) = 9 - 4x^2. We can see that f(-x) = 9 - 4x^2 is equivalent to f(x) = 9 - 4x^2.

Conclusion

Based on the comparison in Step 3, we can conclude that f(-x) = 9 - 4x^2 is indeed equivalent to f(x) = 9 - 4x^2. However, this does not necessarily mean that the function is odd. To determine whether the function is odd, we need to check if f(-x) = -f(x).

Step 4: Check if f(-x) = -f(x)

Let's check if f(-x) = -f(x) for the given function f(x) = 9 - 4x^2. We have already found that f(-x) = 9 - 4x^2. To check if f(-x) = -f(x), we need to multiply f(x) by -1. This gives us -f(x) = -1(9 - 4x^2) = -9 + 4x^2.

Step 5: Compare f(-x) and -f(x)

Now that we have found f(-x) = 9 - 4x^2 and -f(x) = -9 + 4x^2, let's compare them. We can see that f(-x) ≠ -f(x).

Conclusion

Based on the comparison in Step 5, we can conclude that f(-x) ≠ -f(x). Therefore, the function f(x) = 9 - 4x^2 is not an odd function.

Which Statement Best Describes How to Determine Whether a Function is Odd?

Based on our analysis, we can conclude that the correct statement is:

A. Determine whether 9 - 4(-x)^2 is equivalent to 9 - 4x^2.

This statement is correct because it describes the process of substituting -x into the function and simplifying the expression to determine whether the function is odd.

Discussion

In this article, we have explored how to determine whether a function is odd or even. We have used the function f(x) = 9 - 4x^2 as an example and have shown that it is not an odd function. We have also discussed the properties of odd and even functions and have provided a step-by-step guide on how to determine whether a function is odd.

Conclusion

In conclusion, determining whether a function is odd or even is an important concept in mathematics. By following the steps outlined in this article, you can determine whether a given function is odd or even. Remember to substitute -x into the function, simplify the expression, and compare it with the original function to determine whether the function is odd.

References

Frequently Asked Questions

  • 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 the domain of the function.
  • 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 the domain of the function.
  • Q: How do I determine whether a function is odd or even?
  • A: To determine whether a function is odd or even, substitute -x into the function, simplify the expression, and compare it with the original function.
    Q&A: Determining Whether a Function is Odd or Even =====================================================

Introduction

In our previous article, we explored how to determine whether a function is odd or even. We used the function f(x) = 9 - 4x^2 as an example and showed that it is not an odd function. In this article, we will answer some frequently asked questions about determining whether a function is odd or even.

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) for all x in the domain of the function. An even function is a function that satisfies the condition f(-x) = f(x) for all x in the domain of the function.

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

A: To determine whether a function is odd or even, substitute -x into the function, simplify the expression, and compare it with the original function.

Q: What is the significance of determining whether a function is odd or even?

A: Determining whether a function is odd or even is important in mathematics because it helps us understand the behavior of the function. For example, if a function is odd, it means that the function is symmetric about the origin, while if a function is even, it means that the function is symmetric about the y-axis.

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 means that f(-x) = -f(x), while if a function is even, it means that f(-x) = f(x). These two conditions are mutually exclusive, so a function cannot satisfy both conditions.

Q: How do I determine whether a function is odd or even using a graph?

A: To determine whether a function is odd or even using a graph, look for symmetry. If the graph is symmetric about the origin, the function is odd. If the graph is symmetric about the y-axis, the function is even.

Q: Can a function be odd or even if it is not defined at x = 0?

A: Yes, a function can be odd or even even if it is not defined at x = 0. The definition of an odd or even function only requires that the function satisfy the condition f(-x) = -f(x) or f(-x) = f(x) for all x in the domain of the function, except possibly at x = 0.

Q: How do I determine whether a function is odd or even using calculus?

A: To determine whether a function is odd or even using calculus, take the derivative of the function and evaluate it at x = 0. If the derivative is zero at x = 0, the function is even. If the derivative is nonzero at x = 0, the function is odd.

Q: Can a function be odd or even if it is not continuous?

A: Yes, a function can be odd or even even if it is not continuous. The definition of an odd or even function only requires that the function satisfy the condition f(-x) = -f(x) or f(-x) = f(x) for all x in the domain of the function, except possibly at points where the function is not continuous.

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, we can better understand the behavior of functions and make predictions about their behavior. We hope that this article has been helpful in answering your questions about determining whether a function is odd or even.

References

Frequently Asked Questions

  • 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 the domain of the function.
  • 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 the domain of the function.
  • Q: How do I determine whether a function is odd or even?
  • A: To determine whether a function is odd or even, substitute -x into the function, simplify the expression, and compare it with the original function.