To determine whether [tex]\( f(x) = 9 - 4x^2 \)[/tex] is an odd function, we need to verify if [tex]\( f(-x) = -f(x) \)[/tex]. A function [tex]\( f(x) \)[/tex] is considered odd if this condition holds true.
Here is the detailed step-by-step analysis:
1. Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 9 - 4(-x)^2
\][/tex]
Since [tex]\( (-x)^2 = x^2 \)[/tex], this simplifies to:
[tex]\[
f(-x) = 9 - 4x^2
\][/tex]
2. Compute [tex]\(-f(x)\)[/tex]:
[tex]\[
-f(x) = -(9 - 4x^2) = -9 + 4x^2
\][/tex]
3. Verify if [tex]\( f(-x) \)[/tex] is equivalent to [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = 9 - 4x^2 \quad \text{and} \quad -f(x) = -9 + 4x^2
\][/tex]
By comparing [tex]\( f(-x) \)[/tex] and [tex]\(-f(x) \)[/tex], we see that:
[tex]\[
9 - 4x^2 \neq -9 + 4x^2
\][/tex]
Since these two expressions are not equivalent, [tex]\( f(x) = 9 - 4x^2 \)[/tex] is not an odd function.
Therefore, the correct statement to determine if [tex]\( f(x) \)[/tex] is an odd function is:
Determine whether [tex]\( 9 - 4(-x)^2 \)[/tex] is equivalent to [tex]\( -\left(9 - 4x^2\right) \)[/tex].
However, given that we have already found that [tex]\( 9 - 4(-x)^2 \)[/tex] is not equivalent to [tex]\( -\left(9 - 4x^2\right) \)[/tex], we conclude that [tex]\( f(x) \)[/tex] is not an odd function.
