Answer :
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].
