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]9+4 x^2[/tex].
D. Determine whether [tex]9-4(-x^2)[/tex] is equivalent to [tex]-(9+4 x^2)[/tex].

To check if a function is odd, verify if f ( − x ) = − f ( x ) .
Substitute − x into the function: f ( − x ) = 9 − 4 ( − x ) 2 = 9 − 4 x 2 .
Find the negative of the original function: − f ( x ) = − ( 9 − 4 x 2 ) = − 9 + 4 x 2 .
Determine if 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) . The function is odd if they are equal, otherwise it is not. The correct statement is: 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 ) . This means we need to substitute − x into the function and see if the result is equal to the negative of the original function.

Finding f(-x) First, let's find f ( − x ) for the given function f ( x ) = 9 − 4 x 2 . We substitute − x for x :
f ( − x ) = 9 − 4 ( − x ) 2

Simplifying f(-x) Now, let's simplify f ( − x ) :
f ( − x ) = 9 − 4 ( − x ) 2 = 9 − 4 ( x 2 ) = 9 − 4 x 2

Finding -f(x) Next, let's find − f ( x ) :
− f ( x ) = − ( 9 − 4 x 2 ) = − 9 + 4 x 2

Comparing f(-x) and -f(x) Now we need to check if f ( − x ) is equal to − f ( x ) . We found that f ( − x ) = 9 − 4 x 2 and − f ( x ) = − 9 + 4 x 2 . Since 9 − 4 x 2 is not equal to − 9 + 4 x 2 , the function is not odd.

Conclusion The question asks which statement best describes how to determine if f ( x ) is an odd function. We need to determine whether f ( − x ) = 9 − 4 ( − x ) 2 is equivalent to − f ( x ) = − ( 9 − 4 x 2 ) . The correct statement is: Determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) .


Examples
Understanding odd and even functions is useful in physics, especially when dealing with symmetric potentials or systems. For example, in quantum mechanics, the parity of a wavefunction (whether it's even or odd) determines certain properties of the system, such as selection rules for transitions between energy levels. Knowing how to determine if a function is odd or even helps simplify calculations and understand the behavior of physical systems.

Answered by GinnyAnswer | 2025-07-03

To check if the function f ( x ) = 9 − 4 x 2 is an odd function, we compare f ( − x ) and − f ( x ) . We find that f ( − x ) = 9 − 4 x 2 and − f ( x ) = − 9 + 4 x 2 are not equal, thus it is not an odd function. The correct statement is option B: determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) .
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Answered by Anonymous | 2025-07-04