Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
