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

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



Answer :

To determine whether a function [tex]\( f(x) \)[/tex] is an odd function, we need to check if [tex]\( f(-x) = -f(x) \)[/tex].

Let's consider the function [tex]\( f(x) = 9 - 4x^2 \)[/tex].

1. First, we find [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = 9 - 4(-x)^2 \][/tex]
Simplifying inside the parentheses:
[tex]\[ (-x)^2 = x^2, \quad \text{so} \quad f(-x) = 9 - 4x^2 \][/tex]

2. Now, we find [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -(9 - 4x^2) = -9 + 4x^2 \][/tex]

3. We check whether [tex]\( f(-x) \)[/tex] is equal to [tex]\( -f(x) \)[/tex]:
[tex]\[ f(-x) = 9 - 4x^2 \][/tex]
[tex]\[ -f(x) = -9 + 4x^2 \][/tex]
Clearly, [tex]\( 9 - 4x^2 \)[/tex] is not equal to [tex]\( -9 + 4x^2 \)[/tex].

Hence, [tex]\( f(-x) \neq -f(x) \)[/tex], which means that [tex]\( f(x) = 9 - 4x^2 \)[/tex] is not an odd function.

Given the options:
- Determine whether [tex]\( 9 - 4(-x)^2 \)[/tex] is equivalent to [tex]\( -(9 - 4x^2) \)[/tex].

This is the correct statement to check whether the function is odd. Since the function does not satisfy the condition [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) = 9 - 4x^2 \)[/tex] is not an odd function.

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