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].
Answer (2)
To check if f ( x ) is odd, verify if f ( − x ) = − f ( x ) .
Calculate f ( − x ) = 9 − 4 ( − x ) 2 = 9 − 4 x 2 .
Calculate − f ( x ) = − ( 9 − 4 x 2 ) = − 9 + 4 x 2 .
Determine if 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 f ( − x ) and − f ( x ) and see if they are equivalent.
Calculating f(-x) First, let's find f ( − x ) . We substitute − x for x in the function: f ( − x ) = 9 − 4 ( − x ) 2 Since ( − x ) 2 = x 2 , we have: f ( − x ) = 9 − 4 x 2
Calculating -f(x) Next, let's find − f ( x ) . We multiply the entire function by − 1 :
− f ( x ) = − ( 9 − 4 x 2 ) = − 9 + 4 x 2
Checking for Equivalence Now we need to check which of the given statements is the correct way to determine if f ( x ) is odd. We found that f ( − x ) = 9 − 4 x 2 and − f ( x ) = − 9 + 4 x 2 . For f ( x ) to be odd, f ( − x ) must be equal to − f ( x ) . Therefore, we need to check if 9 − 4 x 2 is equivalent to − 9 + 4 x 2 .
Conclusion The statement that best describes how to determine whether f ( x ) = 9 − 4 x 2 is an odd function is: Determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) .
Examples
Understanding odd and even functions helps in analyzing symmetrical patterns in various fields. For instance, in physics, understanding symmetry simplifies the analysis of wave functions. In signal processing, recognizing even or odd signals can optimize data compression and filtering techniques. The concept of symmetry, as explored through odd and even functions, provides a powerful tool for simplifying complex problems across different scientific and engineering disciplines.
To determine if f ( x ) = 9 − 4 x 2 is odd, we check if f ( − x ) = − f ( x ) . Upon calculations, we find that f ( − x ) is not equal to − f ( x ) , meaning the function is not odd. Therefore, option B is the correct choice: to determine whether 9 − 4 ( − x ) 2 is equivalent to − ( 9 − 4 x 2 ) .
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