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.
