Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To analyze the function [tex]\( g(x) = \frac{10}{x} \)[/tex] and determine relevant properties such as domain, range, and whether it increases or decreases, follow these detailed steps:
1. Domain Analysis:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function [tex]\( g(x) = \frac{10}{x} \)[/tex], we need to identify the x-values that would result in a valid output.
- The only restriction here is that the denominator [tex]\( x \)[/tex] cannot be zero because division by zero is undefined.
- Therefore, the domain of [tex]\( g(x) = \frac{10}{x} \)[/tex] is all real numbers except [tex]\( x = 0 \)[/tex]. This can be written as:
[tex]\[\text{Domain} = \{ x \in \mathbb{R} \mid x \neq 0 \}\][/tex]
2. Range Analysis:
- The range of a function is the set of all possible output values (y-values) that the function can take. For the function [tex]\( g(x) = \frac{10}{x} \)[/tex], we need to analyze the y-values obtained as [tex]\( x \)[/tex] approaches all possible values within its domain.
- When [tex]\( x \)[/tex] is a very large positive number or a very large negative number, [tex]\( g(x) \)[/tex] approaches 0 but it can never actually be 0.
- For all other [tex]\( x \)[/tex] (positive and negative), [tex]\( g(x) \)[/tex] can take any positive or negative value.
- Hence, the range of [tex]\( g(x) = \frac{10}{x} \)[/tex] is all real numbers except 0. This is written as:
[tex]\[\text{Range} = \{ y \in \mathbb{R} \mid y \neq 0 \}\][/tex]
3. Behavior Analysis:
- To determine whether the function is increasing or decreasing, let's consider the nature of [tex]\( g(x) = \frac{10}{x} \)[/tex].
- A function is increasing if its derivative [tex]\( g'(x) \)[/tex] is positive and decreasing if [tex]\( g'(x) \)[/tex] is negative. The derivative of [tex]\( g(x) = \frac{10}{x} \)[/tex] is:
[tex]\[ g'(x) = -\frac{10}{x^2} \][/tex]
- Notice that [tex]\( g'(x) \)[/tex] is always negative for all [tex]\( x \neq 0 \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is a strictly decreasing function for its entire domain (excluding [tex]\( x = 0 \)[/tex]).
Based on this analysis, we can select the correct statements:
1. The domain of [tex]\( g(x) \)[/tex] is not all real numbers, since it excludes [tex]\( x = 0 \)[/tex].
2. The domain of [tex]\( g(x) \)[/tex] is not necessarily the same as the domain of a parent function, since the parent function (like [tex]\( f(x) = \frac{1}{x} \)[/tex]) has a similar domain.
3. The range of [tex]\( g(x) \)[/tex] is not all real numbers since it does not include 0.
4. The range is the same as the range of the parent function [tex]\( \frac{1}{x} \)[/tex], excluding 0.
5. The function [tex]\( g(x) \)[/tex] does not increase over any [tex]\( x \)[/tex] values; it strictly decreases.
6. The function [tex]\( g(x) \)[/tex] does decrease over the same [tex]\( x \)[/tex] values as the parent function [tex]\( \frac{1}{x} \)[/tex].
Therefore, the true statements are:
- The domain of [tex]\( g(x) \)[/tex] is the same as the domain of the parent function [tex]\( \frac{1}{x} \)[/tex].
- The range is the same as the range of the parent function [tex]\( \frac{1}{x} \)[/tex].
- The function [tex]\( g(x) \)[/tex] decreases over the same [tex]\( x \)[/tex] values as the parent function [tex]\( \frac{1}{x} \)[/tex].
1. Domain Analysis:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function [tex]\( g(x) = \frac{10}{x} \)[/tex], we need to identify the x-values that would result in a valid output.
- The only restriction here is that the denominator [tex]\( x \)[/tex] cannot be zero because division by zero is undefined.
- Therefore, the domain of [tex]\( g(x) = \frac{10}{x} \)[/tex] is all real numbers except [tex]\( x = 0 \)[/tex]. This can be written as:
[tex]\[\text{Domain} = \{ x \in \mathbb{R} \mid x \neq 0 \}\][/tex]
2. Range Analysis:
- The range of a function is the set of all possible output values (y-values) that the function can take. For the function [tex]\( g(x) = \frac{10}{x} \)[/tex], we need to analyze the y-values obtained as [tex]\( x \)[/tex] approaches all possible values within its domain.
- When [tex]\( x \)[/tex] is a very large positive number or a very large negative number, [tex]\( g(x) \)[/tex] approaches 0 but it can never actually be 0.
- For all other [tex]\( x \)[/tex] (positive and negative), [tex]\( g(x) \)[/tex] can take any positive or negative value.
- Hence, the range of [tex]\( g(x) = \frac{10}{x} \)[/tex] is all real numbers except 0. This is written as:
[tex]\[\text{Range} = \{ y \in \mathbb{R} \mid y \neq 0 \}\][/tex]
3. Behavior Analysis:
- To determine whether the function is increasing or decreasing, let's consider the nature of [tex]\( g(x) = \frac{10}{x} \)[/tex].
- A function is increasing if its derivative [tex]\( g'(x) \)[/tex] is positive and decreasing if [tex]\( g'(x) \)[/tex] is negative. The derivative of [tex]\( g(x) = \frac{10}{x} \)[/tex] is:
[tex]\[ g'(x) = -\frac{10}{x^2} \][/tex]
- Notice that [tex]\( g'(x) \)[/tex] is always negative for all [tex]\( x \neq 0 \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] is a strictly decreasing function for its entire domain (excluding [tex]\( x = 0 \)[/tex]).
Based on this analysis, we can select the correct statements:
1. The domain of [tex]\( g(x) \)[/tex] is not all real numbers, since it excludes [tex]\( x = 0 \)[/tex].
2. The domain of [tex]\( g(x) \)[/tex] is not necessarily the same as the domain of a parent function, since the parent function (like [tex]\( f(x) = \frac{1}{x} \)[/tex]) has a similar domain.
3. The range of [tex]\( g(x) \)[/tex] is not all real numbers since it does not include 0.
4. The range is the same as the range of the parent function [tex]\( \frac{1}{x} \)[/tex], excluding 0.
5. The function [tex]\( g(x) \)[/tex] does not increase over any [tex]\( x \)[/tex] values; it strictly decreases.
6. The function [tex]\( g(x) \)[/tex] does decrease over the same [tex]\( x \)[/tex] values as the parent function [tex]\( \frac{1}{x} \)[/tex].
Therefore, the true statements are:
- The domain of [tex]\( g(x) \)[/tex] is the same as the domain of the parent function [tex]\( \frac{1}{x} \)[/tex].
- The range is the same as the range of the parent function [tex]\( \frac{1}{x} \)[/tex].
- The function [tex]\( g(x) \)[/tex] decreases over the same [tex]\( x \)[/tex] values as the parent function [tex]\( \frac{1}{x} \)[/tex].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
