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].
