To determine which of the given functions are odd, we need to check each function [tex]\( f(x) \)[/tex] to see if it satisfies the condition for an odd function. A function is considered odd if:
[tex]\[
f(-x) = -f(x)
\][/tex]
Let's test each function one by one.
1. Function [tex]\( f(x) = x^3 - x^2 \)[/tex]:
- Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x)^3 - (-x)^2 = -x^3 - x^2
\][/tex]
- Compare [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(x^3 - x^2) = -x^3 + x^2
\][/tex]
Since [tex]\( f(-x) = -x^3 - x^2 \)[/tex] is not equal to [tex]\( -f(x) = -x^3 + x^2 \)[/tex], the function [tex]\( x^3 - x^2 \)[/tex] is not odd.
2. Function [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex]:
- Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x)^5 - 3(-x)^3 + 2(-x) = -x^5 + 3x^3 - 2x
\][/tex]
- Compare [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(x^5 - 3x^3 + 2x) = -x^5 + 3x^3 - 2x
\][/tex]
Since [tex]\( f(-x) = -x^5 + 3x^3 - 2x \)[/tex] is equal to [tex]\( -f(x) = -x^5 + 3x^3 - 2x \)[/tex], the function [tex]\( x^5 - 3x^3 + 2x \)[/tex] is odd.
3. Function [tex]\( f(x) = 4x + 9 \)[/tex]:
- Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 4(-x) + 9 = -4x + 9
\][/tex]
- Compare [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(4x + 9) = -4x - 9
\][/tex]
Since [tex]\( f(-x) = -4x + 9 \)[/tex] is not equal to [tex]\( -f(x) = -4x - 9 \)[/tex], the function [tex]\( 4x + 9 \)[/tex] is not odd.
4. Function [tex]\( f(x) = \frac{1}{x} \)[/tex]:
- Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = \frac{1}{-x} = -\frac{1}{x}
\][/tex]
- Compare [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -\left(\frac{1}{x}\right) = -\frac{1}{x}
\][/tex]
Since [tex]\( f(-x) = -\frac{1}{x} \)[/tex] is equal to [tex]\( -f(x) = -\frac{1}{x} \)[/tex], the function [tex]\( \frac{1}{x} \)[/tex] is odd.
Based on the analysis, the functions that are odd are:
[tex]\[
f(x) = x^5 - 3x^3 + 2x \quad \text{and} \quad f(x) = \frac{1}{x}
\][/tex]
So, the odd functions are:
[tex]\[
\boxed{f(x) = x^5 - 3x^3 + 2x \quad \text{and} \quad f(x) = \frac{1}{x}}
\][/tex]
