atticus30
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Which statement best describes how to determine whether \( f(x) = 9 - 4x^2 \) is an odd function?

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

Sagot :

GinnyAnswer
To determine whether the function \( f(x) = 9 - 4x^2 \) is an odd function, we need to check if \( f(-x) = -f(x) \).

1. First, we evaluate \( f(-x) \):
[tex]\[ f(-x) = 9 - 4(-x)^2 \][/tex]
This simplifies as follows:
[tex]\[ f(-x) = 9 - 4(x^2) \][/tex]
[tex]\[ f(-x) = 9 - 4x^2 \][/tex]

2. Next, we compute \(-f(x)\):
[tex]\[ -f(x) = -(9 - 4x^2) \][/tex]
This simplifies as follows:
[tex]\[ -f(x) = -9 + 4x^2 \][/tex]
[tex]\[ -f(x) = 4x^2 - 9 \][/tex]

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

We observe that:
[tex]\[ f(-x) \neq -f(x) \][/tex]

Since \( f(-x) \) is not equal to \(-f(x) \), the function \( f(x) = 9 - 4x^2 \) is not an odd function.

Thus, the correct statement is:
Determine whether [tex]\( 9 - 4(-x)^2 \)[/tex] is equivalent to [tex]\( -\left( 9 - 4x^2 \right) \)[/tex].
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