To determine whether each function is even, odd, or neither, we need to understand the definitions of even and odd functions:
- A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
- A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
Now, let's analyze each given function step-by-step.
1. Function [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex]
To check whether this function is even, odd, or neither, let's substitute [tex]\( -x \)[/tex] into the function:
[tex]\( f(-x) = \sqrt{(-x)^2} - 9 \)[/tex]
[tex]\( f(-x) = \sqrt{x^2} - 9 \)[/tex]
[tex]\( f(-x) = \sqrt{x^2} - 9 \)[/tex]
As we can see, [tex]\( f(-x) = f(x) \)[/tex]. Therefore, the function [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex] is even.
2. Function [tex]\( g(x) = |x - 3| \)[/tex]
To check whether this function is even, odd, or neither, let's substitute [tex]\( -x \)[/tex] into the function:
[tex]\( g(-x) = |-x - 3| \)[/tex]
[tex]\( g(-x) = |-x - 3| \)[/tex]
We notice that [tex]\( g(-x) \neq g(x) \)[/tex] and [tex]\( g(-x) \neq -g(x) \)[/tex]. Therefore, the function [tex]\( g(x) = |x - 3| \)[/tex] is neither even nor odd.
3. Function [tex]\( f(x) = \frac{x}{x^2-1} \)[/tex]
To check whether this function is even, odd, or neither, let's substitute [tex]\( -x \)[/tex] into the function:
[tex]\( f(-x) = \frac{-x}{(-x)^2-1} \)[/tex]
[tex]\( f(-x) = \frac{-x}{x^2-1} \)[/tex]
We notice that [tex]\( f(-x) = -\frac{x}{x^2-1} = -f(x) \)[/tex]. Therefore, the function [tex]\( f(x) = \frac{x}{x^2-1} \)[/tex] is odd.
4. Function [tex]\( g(x) = x + x^2 \)[/tex]
To check whether this function is even, odd, or neither, let's substitute [tex]\( -x \)[/tex] into the function:
[tex]\( g(-x) = -x + (-x)^2 \)[/tex]
[tex]\( g(-x) = -x + x^2 \)[/tex]
We notice that [tex]\( g(-x) \neq g(x) \)[/tex] and [tex]\( g(-x) \neq -g(x) \)[/tex]. Therefore, the function [tex]\( g(x) = x + x^2 \)[/tex] is neither even nor odd.
In conclusion, the results are:
- [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex] is even.
- [tex]\( g(x) = |x - 3| \)[/tex] is neither.
- [tex]\( f(x) = \frac{x}{x^2-1} \)[/tex] is odd.
- [tex]\( g(x) = x + x^2 \)[/tex] is neither.
