To determine which functions are odd, we need to recall the definition of an odd function. A function [tex]\( f(x) \)[/tex] is considered odd if it satisfies the condition:
[tex]\[ f(-x) = -f(x) \][/tex]
Let's examine each function one by one.
### 1. [tex]\( f(x) = x^3 - x^2 \)[/tex]
Firstly, we find [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^3 - (-x)^2 = -x^3 - x^2 \][/tex]
To check if [tex]\( f(x) \)[/tex] is odd, we compare [tex]\( f(-x) \)[/tex] with [tex]\(-f(x)\)[/tex]:
[tex]\[ -f(x) = -(x^3 - x^2) = -x^3 + x^2 \][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) = x^3 - x^2 \)[/tex] is not odd.
### 2. [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex]
Now consider [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^5 - 3(-x)^3 + 2(-x) = -x^5 + 3x^3 - 2x \][/tex]
We compare this with [tex]\(-f(x)\)[/tex]:
[tex]\[ -f(x) = -(x^5 - 3x^3 + 2x) = -x^5 + 3x^3 - 2x \][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex] is an odd function.
### 3. [tex]\( f(x) = 4x + 9 \)[/tex]
Now, let's evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = 4(-x) + 9 = -4x + 9 \][/tex]
To check the condition for being odd, we compare this with [tex]\(-f(x)\)[/tex]:
[tex]\[ -f(x) = -(4x + 9) = -4x - 9 \][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) = 4x + 9 \)[/tex] is not odd.
### 4. [tex]\( f(x) = \frac{1}{x} \)[/tex]
Finally, consider [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = \frac{1}{-x} = -\frac{1}{x} \][/tex]
We compare this with [tex]\(-f(x)\)[/tex]:
[tex]\[ -f(x) = -\left(\frac{1}{x}\right) = -\frac{1}{x} \][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) = \frac{1}{x} \)[/tex] is an odd function.
### Conclusion:
The functions that are odd among the given list are:
[tex]\[ f(x) = x^5 - 3x^3 + 2x \][/tex]
[tex]\[ f(x) = \frac{1}{x} \][/tex]
