To determine whether each function is even, odd, or neither, we use the definitions of even and odd functions:
1. Even Function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
2. Odd Function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
Let's analyze each function step-by-step.
### 1. [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex]
We need to determine [tex]\( f(-x) \)[/tex].
[tex]\[ f(-x) = \sqrt{(-x)^2} - 9 = \sqrt{x^2} - 9 = f(x) \][/tex]
Since [tex]\( f(-x) = f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is even.
### 2. [tex]\( g(x) = |x - 3| \)[/tex]
We need to determine [tex]\( g(-x) \)[/tex].
[tex]\[ g(-x) = |-x - 3| \][/tex]
To analyze further, let’s see what happens for specific values:
- For [tex]\( x = 3 \)[/tex], [tex]\( g(3) = |3 - 3| = 0 \)[/tex] and [tex]\( g(-3) = |-3 - 3| = |-6| = 6 \)[/tex].
- For [tex]\( x = -3 \)[/tex], [tex]\( g(-3) = |-3 - 3| = 6 \)[/tex] which is not equal to [tex]\( |3 - 3| = 0 \)[/tex].
In general, [tex]\( |x - 3| \neq |-(x - 3)| \)[/tex] unless [tex]\( x = 3 \)[/tex].
Therefore, [tex]\( g(x) \)[/tex] is neither even nor odd.
### 3. [tex]\( f(x) = \frac{x}{x^2 - 1} \)[/tex]
We need to determine [tex]\( f(-x) \)[/tex].
[tex]\[ f(-x) = \frac{-x}{(-x)^2 - 1} = \frac{-x}{x^2 - 1} = -\frac{x}{x^2 - 1} = -f(x) \][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is odd.
### 4. [tex]\( g(x) = x + x^2 \)[/tex]
We need to determine [tex]\( g(-x) \)[/tex].
[tex]\[ g(-x) = -x + (-x)^2 = -x + x^2 \][/tex]
To compare it with [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x + x^2 \][/tex]
[tex]\[ g(-x) = -x + x^2 \][/tex]
Since [tex]\( g(-x) \neq g(x) \)[/tex] and [tex]\( g(-x) \neq -g(x) \)[/tex], the function [tex]\( g(x) \)[/tex] is neither even nor odd.
Therefore, the functions are classified as follows:
1. [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex] is even.
2. [tex]\( g(x) = |x - 3| \)[/tex] is neither.
3. [tex]\( f(x) = \frac{x}{x^2 - 1} \)[/tex] is odd.
4. [tex]\( g(x) = x + x^2 \)[/tex] is neither.
