To determine whether each given function is an odd function, we need to use the definition of odd functions. A function [tex]\( f(x) \)[/tex] is considered odd if [tex]\( f(-x) = -f(x) \)[/tex] for all values of [tex]\( x \)[/tex]. Let's analyze each function one-by-one.
1. For [tex]\( f(x) = -x^2 + 3x - 2 \)[/tex]:
[tex]\[
f(-x) = -(-x)^2 + 3(-x) - 2 = -x^2 - 3x - 2
\][/tex]
[tex]\[
-f(x) = -(-x^2 + 3x - 2) = x^2 - 3x + 2
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] and [tex]\( -f(x) \)[/tex]:
[tex]\[
-x^2 - 3x - 2 \neq x^2 - 3x + 2
\][/tex]
Therefore, [tex]\( f(x) = -x^2 + 3x - 2 \)[/tex] is not an odd function.
2. For [tex]\( f(x) = -3x \)[/tex]:
[tex]\[
f(-x) = -3(-x) = 3x
\][/tex]
[tex]\[
-f(x) = -(-3x) = 3x
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] and [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = 3x = -f(x)
\][/tex]
Therefore, [tex]\( f(x) = -3x \)[/tex] is an odd function.
3. For [tex]\( f(x) = 2x^3 + x \)[/tex]:
[tex]\[
f(-x) = 2(-x)^3 + (-x) = -2x^3 - x
\][/tex]
[tex]\[
-f(x) = -(2x^3 + x) = -2x^3 - x
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] and [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = -2x^3 - x = -f(x)
\][/tex]
Therefore, [tex]\( f(x) = 2x^3 + x \)[/tex] is an odd function.
4. For [tex]\( f(x) = -9x \)[/tex]:
[tex]\[
f(-x) = -9(-x) = 9x
\][/tex]
[tex]\[
-f(x) = -(-9x) = 9x
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] and [tex]\( -f(x) \)[/tex]:
[tex]\[
f(-x) = 9x = -f(x)
\][/tex]
Therefore, [tex]\( f(x) = -9x \)[/tex] is an odd function.
5. For [tex]\( f(x) = -4x + 1 \)[/tex]:
[tex]\[
f(-x) = -4(-x) + 1 = 4x + 1
\][/tex]
[tex]\[
-f(x) = -(-4x + 1) = 4x - 1
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] and [tex]\( -f(x) \)[/tex]:
[tex]\[
4x + 1 \neq 4x - 1
\][/tex]
Therefore, [tex]\( f(x) = -4x + 1 \)[/tex] is not an odd function.
Based on the above analysis, the odd functions among the given options are:
- [tex]\( f(x) = -3x \)[/tex]
- [tex]\( f(x) = 2x^3 + x \)[/tex]
- [tex]\( f(x) = -9x \)[/tex]
So, the correct answers are:
[tex]\( f(x) = -3x \)[/tex], [tex]\( f(x) = 2x^3 + x \)[/tex], and [tex]\( f(x) = -9x \)[/tex].
