Which statement best describes how to determine whether [tex]f(x)=9-4 x^2[/tex] is an odd function?

A. Determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]9-4 x^2[/tex].
B. Determine whether [tex]9-4(-x^2)[/tex] is equivalent to [tex]9+4 x^2[/tex].
C. Determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]-\left(9-4 x^2\right)[/tex].
D. Determine whether [tex]9-4(-x^2)[/tex] is equivalent to [tex]-\left(9+4 x^2\right)[/tex].

To check if a function f ( x ) is odd, verify if f ( − x ) = − f ( x ) .
Calculate f ( − x ) by substituting − x into the function: f ( − x ) = 9 − 4 ( − x ) 2 = 9 − 4 x 2 .
Calculate − f ( x ) by multiplying the function by − 1 : − f ( x ) = − ( 9 − 4 x 2 ) = − 9 + 4 x 2 .
Determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) to check if the function is odd. Determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) ​

Explanation

Understanding Odd Functions To determine if a function f ( x ) is odd, we need to check if f ( − x ) = − f ( x ) . In this case, f ( x ) = 9 − 4 x 2 . We need to find the statement that correctly describes this check.

Calculating f(-x) First, let's find f ( − x ) by substituting − x for x in the function: f ( − x ) = 9 − 4 ( − x ) 2 = 9 − 4 ( x 2 ) = 9 − 4 x 2 So, f ( − x ) = 9 − 4 x 2 .

Calculating -f(x) Next, let's find − f ( x ) by multiplying the function by − 1 :
− f ( x ) = − ( 9 − 4 x 2 ) = − 9 + 4 x 2 So, − f ( x ) = − 9 + 4 x 2 .

Checking the Odd Function Condition Now, we need to check if f ( − x ) = − f ( x ) . In our case, we have: f ( − x ) = 9 − 4 x 2 − f ( x ) = − 9 + 4 x 2 Since 9 − 4 x 2 e q − 9 + 4 x 2 , the function is not odd. However, the question asks which statement best describes how to determine if the function is odd. We need to find the option that correctly sets up the check.

Analyzing the Options Let's analyze the given options:

Determine whether 9 − 4 ( − x ) 2 is equivalent to 9 − 4 x 2 . This is equivalent to checking if f ( − x ) = f ( x ) , which would determine if the function is even, not odd.

Determine whether 9 − 4 ( − x 2 ) is equivalent to 9 + 4 x 2 . This is not a correct way to check for odd or even functions.

Determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) . This is equivalent to checking if f ( − x ) = − f ( x ) , which is the correct condition for an odd function.

Determine whether 9 − 4 ( − x 2 ) is equivalent to − ( 9 + 4 x 2 ) . This is not a correct way to check for odd or even functions.

Final Answer The correct statement is: Determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) .
This is because it checks if f ( − x ) = − f ( x ) , which is the condition for a function to be odd.


Examples
In physics, understanding whether a function is odd or even can simplify the analysis of symmetrical systems. For example, when analyzing the motion of a pendulum, if the restoring force is an odd function of the displacement, it indicates a certain type of symmetry in the system. Similarly, in signal processing, even and odd functions are used to decompose signals into symmetrical components, which can help in filtering and analyzing the signal's frequency content. Identifying these symmetries through odd and even functions allows engineers and physicists to make predictions and simplify complex calculations, leading to more efficient designs and a deeper understanding of the underlying phenomena.

Answered by GinnyAnswer | 2025-07-03

To check if the function f ( x ) = 9 − 4 x 2 is odd, we confirm if f ( − x ) = − f ( x ) . The calculations show that f ( − x ) = 9 − 4 x 2 and − f ( x ) = − 9 + 4 x 2 , establishing that it is not odd. The correct option for determining the function's oddness is option C: check if 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) .
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Answered by Anonymous | 2025-07-04