Sure, let's check each of these functions to determine if they are odd. Recall that a function [tex]\( f(x) \)[/tex] is defined as odd if it satisfies the condition:
[tex]\[ f(-x) = -f(x) \][/tex]
Let's evaluate each function step-by-step to see if they meet this criterion.
### Function 1: [tex]\( f(x) = x^3 - x^2 \)[/tex]
1. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^3 - (-x)^2 = -x^3 - x^2 \][/tex]
2. 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.
### Function 2: [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex]
1. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^5 - 3(-x)^3 + 2(-x) = -x^5 + 3x^3 - 2x \][/tex]
2. 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]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex] is odd.
### Function 3: [tex]\( f(x) = 4x + 9 \)[/tex]
1. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = 4(-x) + 9 = -4x + 9 \][/tex]
2. Compare [tex]\( f(-x) \)[/tex] 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.
### Function 4: [tex]\( f(x) = \frac{1}{x} \)[/tex]
1. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = \frac{1}{-x} = -\frac{1}{x} \][/tex]
2. Compare [tex]\( f(-x) \)[/tex] with [tex]\(-f(x)\)[/tex]:
[tex]\[ -f(x) = -\frac{1}{x} \][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) = \frac{1}{x} \)[/tex] is odd.
### Summary
Based on the steps above, we have determined which functions are odd:
- [tex]\( f(x) = x^3 - x^2 \)[/tex] is not odd.
- [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex] is odd.
- [tex]\( f(x) = 4x + 9 \)[/tex] is not odd.
- [tex]\( f(x) = \frac{1}{x} \)[/tex] is odd.
Thus, the functions that are odd are [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex] and [tex]\( f(x) = \frac{1}{x} \)[/tex].
