To determine which of the given functions are odd, we need to understand the properties of an odd function. A function [tex]\( f(x) \)[/tex] is considered odd if it satisfies the following condition:
[tex]\[ f(-x) = -f(x) \][/tex]
Let's check each given function one by one to see if it meets this criterion.
1. Function [tex]\( g(x) = x^2 \)[/tex]
- Calculate [tex]\( g(-x) \)[/tex]:
[tex]\[ g(-x) = (-x)^2 = x^2 \][/tex]
- Compare [tex]\( g(-x) \)[/tex] with [tex]\(-g(x)\)[/tex]:
[tex]\[ -g(x) = -(x^2) = -x^2 \][/tex]
- Since [tex]\( g(-x) \neq -g(x) \)[/tex]:
[tex]\[ x^2 \neq -x^2 \][/tex]
This function is not odd.
2. Function [tex]\( g(x) = 5x - 1 \)[/tex]
- Calculate [tex]\( g(-x) \)[/tex]:
[tex]\[ g(-x) = 5(-x) - 1 = -5x - 1 \][/tex]
- Compare [tex]\( g(-x) \)[/tex] with [tex]\(-g(x)\)[/tex]:
[tex]\[ -g(x) = -(5x - 1) = -5x + 1 \][/tex]
- Since [tex]\( g(-x) \neq -g(x) \)[/tex]:
[tex]\[ -5x - 1 \neq -5x + 1 \][/tex]
This function is not odd.
3. Function [tex]\( g(x) = 3 \)[/tex]
- Calculate [tex]\( g(-x) \)[/tex]:
[tex]\[ g(-x) = 3 \][/tex]
- Compare [tex]\( g(-x) \)[/tex] with [tex]\(-g(x)\)[/tex]:
[tex]\[ -g(x) = -3 \][/tex]
- Since [tex]\( g(-x) \neq -g(x) \)[/tex]:
[tex]\[ 3 \neq -3 \][/tex]
This function is not odd.
4. Function [tex]\( g(x) = 4x \)[/tex]
- Calculate [tex]\( g(-x) \)[/tex]:
[tex]\[ g(-x) = 4(-x) = -4x \][/tex]
- Compare [tex]\( g(-x) \)[/tex] with [tex]\(-g(x)\)[/tex]:
[tex]\[ -g(x) = -(4x) = -4x \][/tex]
- Since [tex]\( g(-x) = -g(x) \)[/tex]:
[tex]\[ -4x = -4x \][/tex]
This function satisfies the condition for being odd.
Hence, the function that is an odd function is [tex]\( g(x) = 4x \)[/tex].
