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Which of the following are odd functions?

Select all that apply:

A. [tex]\( f(x) = -x^2 + 3x - 2 \)[/tex]
B. [tex]\( f(x) = -3x \)[/tex]
C. [tex]\( f(x) = 2x^3 + x \)[/tex]
D. [tex]\( f(x) = -9x \)[/tex]
E. [tex]\( f(x) = -4x + 1 \)[/tex]

Sagot :

GinnyAnswer
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].
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