Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine which of the given functions is an odd function, we need to verify whether each function satisfies the condition for oddness. A function [tex]\( f(x) \)[/tex] is odd if and only if [tex]\( f(-x) = -f(x) \)[/tex].
Let's go through each function one by one.
1. Function: [tex]\( f(x) = x^3 + 5x^2 + x \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^3 + 5(-x)^2 + (-x) = -x^3 + 5x^2 - x \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -(x^3 + 5x^2 + x) = -x^3 - 5x^2 - x \][/tex]
- Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is not odd.
2. Function: [tex]\( f(x) = \sqrt{x} \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = \sqrt{-x} \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -\sqrt{x} \][/tex]
- Since [tex]\( f(-x) \neq -f(x) \)[/tex] (also note that [tex]\( \sqrt{-x} \)[/tex] is not defined for all real numbers [tex]\( x \)[/tex]), the function [tex]\( f(x) \)[/tex] is not odd.
3. Function: [tex]\( f(x) = x^2 + x \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^2 + (-x) = x^2 - x \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -(x^2 + x) = -x^2 - x \][/tex]
- Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is not odd.
4. Function: [tex]\( f(x) = -x \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = -(-x) = x \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -(-x) = x \][/tex]
- Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is odd.
Based on the analysis, the only odd function among the given functions is [tex]\( f(x) = -x \)[/tex].
Let's go through each function one by one.
1. Function: [tex]\( f(x) = x^3 + 5x^2 + x \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^3 + 5(-x)^2 + (-x) = -x^3 + 5x^2 - x \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -(x^3 + 5x^2 + x) = -x^3 - 5x^2 - x \][/tex]
- Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is not odd.
2. Function: [tex]\( f(x) = \sqrt{x} \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = \sqrt{-x} \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -\sqrt{x} \][/tex]
- Since [tex]\( f(-x) \neq -f(x) \)[/tex] (also note that [tex]\( \sqrt{-x} \)[/tex] is not defined for all real numbers [tex]\( x \)[/tex]), the function [tex]\( f(x) \)[/tex] is not odd.
3. Function: [tex]\( f(x) = x^2 + x \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^2 + (-x) = x^2 - x \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -(x^2 + x) = -x^2 - x \][/tex]
- Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is not odd.
4. Function: [tex]\( f(x) = -x \)[/tex]
- Finding [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = -(-x) = x \][/tex]
- Comparing [tex]\( f(-x) \)[/tex] with [tex]\( -f(x) \)[/tex]:
[tex]\[ -f(x) = -(-x) = x \][/tex]
- Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] is odd.
Based on the analysis, the only odd function among the given functions is [tex]\( f(x) = -x \)[/tex].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.