To determine whether each function is odd, we can use the definition of an odd function. A function [tex]\( f(x) \)[/tex] is considered odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex].
Let's check each function step-by-step:
1. Function [tex]\( f(x) = x^3 - x^2 \)[/tex]:
Calculate [tex]\( f(-x) \)[/tex] and compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = (-x)^3 - (-x)^2 = -x^3 - x^2
\][/tex]
[tex]\[
-f(x) = -(x^3 - x^2) = -x^3 + x^2
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex] (because [tex]\( -x^3 - x^2 \neq -x^3 + x^2 \)[/tex]), the function [tex]\( f(x) = x^3 - x^2 \)[/tex] is not odd.
2. Function [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex]:
Calculate [tex]\( f(-x) \)[/tex] and compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = (-x)^5 - 3(-x)^3 + 2(-x) = -x^5 + 3x^3 - 2x
\][/tex]
[tex]\[
-f(x) = -(x^5 - 3x^3 + 2x) = -x^5 + 3x^3 - 2x
\][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex] (because [tex]\( -x^5 + 3x^3 - 2x = -x^5 + 3x^3 - 2x \)[/tex]), the function [tex]\( f(x) = x^5 - 3x^3 + 2x \)[/tex] is odd.
3. Function [tex]\( f(x) = 4x + 9 \)[/tex]:
Calculate [tex]\( f(-x) \)[/tex] and compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = 4(-x) + 9 = -4x + 9
\][/tex]
[tex]\[
-f(x) = -(4x + 9) = -4x - 9
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex] (because [tex]\( -4x + 9 \neq -4x - 9 \)[/tex]), the function [tex]\( f(x) = 4x + 9 \)[/tex] is not odd.
4. Function [tex]\( f(x) = \frac{1}{x} \)[/tex]:
Calculate [tex]\( f(-x) \)[/tex] and compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = \frac{1}{-x} = -\frac{1}{x}
\][/tex]
[tex]\[
-f(x) = -\left( \frac{1}{x} \right) = -\frac{1}{x}
\][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex] (because [tex]\( -\frac{1}{x} = -\frac{1}{x} \)[/tex]), the function [tex]\( f(x) = \frac{1}{x} \)[/tex] is odd.
Therefore, the odd functions among the given list are:
[tex]\[ f(x) = x^5 - 3x^3 + 2x \][/tex]
and
[tex]\[ f(x) = \frac{1}{x} \][/tex]
This matches the results [tex]\( (False, True, False, True) \)[/tex] for the respective functions.
