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].
Answer (2)
Evaluate f ( − x ) to determine the function's symmetry.
If m is even, f ( − x ) = f ( x ) , indicating an even function.
If m is odd, f ( − x ) e q f ( x ) and f ( − x ) e q − f ( x ) , indicating neither even nor odd function.
Therefore, f ( x ) is an even function for all even values of m , so the answer is: f ( x ) is an even function for all even values of m .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x m + 9 ) 2 and we need to determine the condition on m for which f ( x ) is even or odd. Recall that a function is even if f ( − x ) = f ( x ) and odd if f ( − x ) = − f ( x ) .
Evaluating f(-x) Let's evaluate f ( − x ) : f ( − x ) = (( − x ) m + 9 ) 2
Case: m is even If m is even, then ( − x ) m = x m , so f ( − x ) = ( x m + 9 ) 2 = f ( x ) In this case, f ( x ) is even.
Case: m is odd If m is odd, then ( − x ) m = − x m , so f ( − x ) = ( − x m + 9 ) 2 = ( 9 − x m ) 2 This is not equal to f ( x ) = ( x m + 9 ) 2 or − f ( x ) = − ( x m + 9 ) 2 in general. For example, if x = 1 and m = 1 , then f ( x ) = ( 1 + 9 ) 2 = 100 and f ( − x ) = ( − 1 + 9 ) 2 = 64 . Thus, f ( x ) is neither even nor odd.
Conclusion Therefore, f ( x ) is an even function for all even values of m .
Examples
Understanding even and odd functions is crucial in physics, especially when dealing with symmetrical systems. For instance, the potential energy in a simple harmonic oscillator is an even function, reflecting the symmetry of the system around the equilibrium point. Similarly, analyzing the symmetry of functions helps simplify complex calculations in signal processing and quantum mechanics, making it easier to predict the behavior of physical systems.
The function f ( x ) = ( x m + 9 ) 2 is even when m is even, since f ( − x ) = f ( x ) . When m is odd, f ( − x ) does not equal f ( x ) or − f ( x ) , indicating that it is neither even nor odd. Therefore, the correct choice is option B.
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