If [tex]f(x)=(x^m+9)^2[/tex], which statement about [tex]f(x)[/tex] is true?

A. [tex]f(x)[/tex] is an even function for all values of [tex]m[/tex].
B. [tex]f(x)[/tex] is an even function for all even values of [tex]m[/tex].
C. [tex]f(x)[/tex] is an odd function for all values of [tex]m[/tex].
D. [tex]f(x)[/tex] is an odd function for all odd values of [tex]m[/tex].

Calculate f ( − x ) to determine if f ( x ) is even or odd.
If m is even, f ( − x ) = f ( x ) , so f ( x ) is even.
If m is odd, f ( − x ) e q f ( x ) and f ( − x ) e q − f ( x ) , so f ( x ) is neither even nor odd.
Therefore, f ( x ) is an even function for all even values of m . f ( x ) is an even function for all even values of m ​

Explanation

Understanding Even and Odd Functions We are given the function f ( x ) = ( x m + 9 ) 2 and need to determine for which values of m the function is even or odd. Recall that a function is even if f ( − x ) = f ( x ) for all x , and a function is odd if f ( − x ) = − f ( x ) for all x .

Computing f(-x) Let's compute f ( − x ) : f ( − x ) = (( − x ) m + 9 ) 2

Case: m is even Now, let's consider the case when m is even. If m is even, then ( − x ) m = x m , so f ( − x ) = ( x m + 9 ) 2 = f ( x ) Thus, f ( x ) is an even function when m is even.

Case: m is odd Next, let's consider the case when m is odd. If m is odd, then ( − x ) m = − x m , so f ( − x ) = ( − x m + 9 ) 2 = ( 9 − x m ) 2 In general, ( 9 − x m ) 2 e q ( x m + 9 ) 2 and ( 9 − x m ) 2 e q − ( x m + 9 ) 2 . For example, if m = 1 , then f ( x ) = ( x + 9 ) 2 and f ( − x ) = ( − x + 9 ) 2 . If x = 1 , f ( 1 ) = ( 1 + 9 ) 2 = 100 and f ( − 1 ) = ( − 1 + 9 ) 2 = 64 . Since f ( 1 ) e q f ( − 1 ) and f ( 1 ) e q − f ( − 1 ) , f ( x ) is neither even nor odd when m = 1 .

Conclusion Therefore, f ( x ) is an even function for all even values of m .


Examples
Understanding even and odd functions is useful in physics, especially when dealing with symmetric potentials or symmetric 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 whether a function is even or odd simplifies calculations and provides insights into the behavior of physical systems.

Answered by GinnyAnswer | 2025-07-03

The function f ( x ) = ( x m + 9 ) 2 is even for all even values of m because f ( − x ) = f ( x ) . It is neither even nor odd for odd values of m . Thus, the correct option is B.
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Answered by Anonymous | 2025-07-04