Answer :
Let's determine whether each function is even, odd, or neither.
### 1. [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex]:
1. Rewrite the function:
[tex]\[ f(x) = |x| - 9 \][/tex]
2. Test for evenness: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(x) = f(-x) \)[/tex].
[tex]\[ f(-x) = | -x | - 9 = |x| - 9 = f(x) \][/tex]
Since [tex]\( f(x) = f(-x) \)[/tex], the function [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex] is even.
### 2. [tex]\( g(x) = |x - 3| \)[/tex]:
1. Test for evenness:
[tex]\[ g(x) = |x - 3| \][/tex]
[tex]\[ g(-x) = |-x - 3| \][/tex]
2. Compare [tex]\( g(x) \)[/tex] and [tex]\( g(-x) \)[/tex]:
In general, [tex]\(|x - 3| \neq |-x - 3|\)[/tex], unless [tex]\( x = 3 \)[/tex] or [tex]\( x = -3 \)[/tex]
Since [tex]\( g(x) \neq g(-x) \)[/tex], this function is not even.
3. Test for oddness: A function [tex]\( g(x) \)[/tex] is odd if [tex]\( g(x) = -g(-x) \)[/tex].
[tex]\[ -g(-x) = -|-x - 3| \][/tex]
In general, [tex]\( |x - 3| \neq -|-x - 3| \)[/tex]
Since [tex]\( g(x) \neq -g(-x) \)[/tex] either, the function [tex]\( g(x) = |x - 3| \)[/tex] is neither even nor odd.
### 3. [tex]\( f(x) = \frac{x}{x^2 - 1} \)[/tex]:
1. Test for evenness:
[tex]\[ f(-x) = \frac{-x}{(-x)^2 - 1} = \frac{-x}{x^2 - 1} \][/tex]
Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = -f(x) \][/tex]
Since [tex]\( f(x) \neq f(-x) \)[/tex], this function is not even.
2. Test for oddness:
[tex]\[ f(-x) = -f(x) \][/tex]
Since [tex]\( f(x) = -f(-x) \)[/tex], the function [tex]\( f(x) = \frac{x}{x^2 - 1} \)[/tex] is odd.
### 4. [tex]\( g(x) = x + x^2 \)[/tex]:
1. Test for evenness:
[tex]\[ g(x) = x + x^2 \][/tex]
[tex]\[ g(-x) = -x + (-x)^2 = -x + x^2 \][/tex]
Compare [tex]\( g(x) \)[/tex] and [tex]\( g(-x) \)[/tex]:
[tex]\[ g(x) = x + x^2 \neq -x + x^2 = g(-x) \][/tex]
Since [tex]\( g(x) \neq g(-x) \)[/tex], the function is not even.
2. Test for oddness:
[tex]\[ g(x) = x + x^2 \][/tex]
[tex]\[ -g(-x) = -(-x + x^2) = x - x^2 \][/tex]
Compare [tex]\( g(x) \)[/tex] and [tex]\( -g(-x) \)[/tex]:
[tex]\[ x + x^2 \neq -(-x + x^2) = x - x^2 \][/tex]
Since [tex]\( g(x) \neq -g(-x) \)[/tex], the function [tex]\( g(x) = x + x^2 \)[/tex] is neither even nor odd.
### Final Results:
- [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.
### 1. [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex]:
1. Rewrite the function:
[tex]\[ f(x) = |x| - 9 \][/tex]
2. Test for evenness: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(x) = f(-x) \)[/tex].
[tex]\[ f(-x) = | -x | - 9 = |x| - 9 = f(x) \][/tex]
Since [tex]\( f(x) = f(-x) \)[/tex], the function [tex]\( f(x) = \sqrt{x^2} - 9 \)[/tex] is even.
### 2. [tex]\( g(x) = |x - 3| \)[/tex]:
1. Test for evenness:
[tex]\[ g(x) = |x - 3| \][/tex]
[tex]\[ g(-x) = |-x - 3| \][/tex]
2. Compare [tex]\( g(x) \)[/tex] and [tex]\( g(-x) \)[/tex]:
In general, [tex]\(|x - 3| \neq |-x - 3|\)[/tex], unless [tex]\( x = 3 \)[/tex] or [tex]\( x = -3 \)[/tex]
Since [tex]\( g(x) \neq g(-x) \)[/tex], this function is not even.
3. Test for oddness: A function [tex]\( g(x) \)[/tex] is odd if [tex]\( g(x) = -g(-x) \)[/tex].
[tex]\[ -g(-x) = -|-x - 3| \][/tex]
In general, [tex]\( |x - 3| \neq -|-x - 3| \)[/tex]
Since [tex]\( g(x) \neq -g(-x) \)[/tex] either, the function [tex]\( g(x) = |x - 3| \)[/tex] is neither even nor odd.
### 3. [tex]\( f(x) = \frac{x}{x^2 - 1} \)[/tex]:
1. Test for evenness:
[tex]\[ f(-x) = \frac{-x}{(-x)^2 - 1} = \frac{-x}{x^2 - 1} \][/tex]
Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = -f(x) \][/tex]
Since [tex]\( f(x) \neq f(-x) \)[/tex], this function is not even.
2. Test for oddness:
[tex]\[ f(-x) = -f(x) \][/tex]
Since [tex]\( f(x) = -f(-x) \)[/tex], the function [tex]\( f(x) = \frac{x}{x^2 - 1} \)[/tex] is odd.
### 4. [tex]\( g(x) = x + x^2 \)[/tex]:
1. Test for evenness:
[tex]\[ g(x) = x + x^2 \][/tex]
[tex]\[ g(-x) = -x + (-x)^2 = -x + x^2 \][/tex]
Compare [tex]\( g(x) \)[/tex] and [tex]\( g(-x) \)[/tex]:
[tex]\[ g(x) = x + x^2 \neq -x + x^2 = g(-x) \][/tex]
Since [tex]\( g(x) \neq g(-x) \)[/tex], the function is not even.
2. Test for oddness:
[tex]\[ g(x) = x + x^2 \][/tex]
[tex]\[ -g(-x) = -(-x + x^2) = x - x^2 \][/tex]
Compare [tex]\( g(x) \)[/tex] and [tex]\( -g(-x) \)[/tex]:
[tex]\[ x + x^2 \neq -(-x + x^2) = x - x^2 \][/tex]
Since [tex]\( g(x) \neq -g(-x) \)[/tex], the function [tex]\( g(x) = x + x^2 \)[/tex] is neither even nor odd.
### Final Results:
- [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.
