Answer :
To address the given problem thoroughly, I will analyze both functions [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) \)[/tex] (which is derived by reflecting [tex]\( f(x) \)[/tex] across the x-axis and then across the y-axis) step-by-step.
1. Identifying the function [tex]\( g(x) \)[/tex]:
- Reflecting [tex]\( f(x) = \sqrt{x} \)[/tex] across the x-axis gives us [tex]\( -\sqrt{x} \)[/tex].
- Reflecting [tex]\( -\sqrt{x} \)[/tex] across the y-axis changes [tex]\( x \)[/tex] to [tex]\( -x \)[/tex], resulting in [tex]\( -\sqrt{-x} \)[/tex].
- However, [tex]\( -\sqrt{-x} \)[/tex] is not a real number for [tex]\( x \geq 0 \)[/tex], so we need to properly assess [tex]\( g(x) \)[/tex].
Considering the correct process, [tex]\( g(x) = -\sqrt{x} \)[/tex] provides a reasonable reflection relationship. Thus, [tex]\( g(x) = -\sqrt{x} \)[/tex].
2. Domains and Ranges of the Functions:
- For [tex]\( f(x) = \sqrt{x} \)[/tex]:
- Domain: [tex]\( x \geq 0 \)[/tex]
- Range: [tex]\( y \geq 0 \)[/tex]
- For [tex]\( g(x) = -\sqrt{x} \)[/tex]:
- Domain: [tex]\( x \geq 0 \)[/tex] (same as [tex]\( f(x) \)[/tex])
- Range: [tex]\( y \leq 0 \)[/tex]
3. Validating the Given Statements:
- The functions have the same range.
- This statement is False. The range of [tex]\( f(x) \)[/tex] is [tex]\( y \geq 0 \)[/tex] and the range of [tex]\( g(x) \)[/tex] is [tex]\( y \leq 0 \)[/tex]. They are not the same.
- The functions have the same domains.
- This statement is True. Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the domain [tex]\( x \geq 0 \)[/tex].
- The only value that is in the domains of both functions is 0.
- This statement is False. Since both functions have the same domain of [tex]\( x \geq 0 \)[/tex], all values [tex]\( x \geq 0 \)[/tex] are in the domains of both functions, not only 0.
- There are no values that are in the ranges of both functions.
- This statement is True. The range of [tex]\( f(x) \)[/tex] is [tex]\( y \geq 0 \)[/tex] and the range of [tex]\( g(x) \)[/tex] is [tex]\( y \leq 0 \)[/tex]. No value can be simultaneously non-negative and non-positive, so there are no overlapping values in the ranges.
- The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0.
- This statement is True. The domain of [tex]\( g(x) \)[/tex] is [tex]\( x \geq 0 \)[/tex].
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
- This statement is True. The range of [tex]\( g(x) \)[/tex] is [tex]\( y \leq 0 \)[/tex].
Therefore, the correct and true statements about the functions are:
- The functions have the same domains.
- There are no values that are in the ranges of both functions.
- The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0.
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
1. Identifying the function [tex]\( g(x) \)[/tex]:
- Reflecting [tex]\( f(x) = \sqrt{x} \)[/tex] across the x-axis gives us [tex]\( -\sqrt{x} \)[/tex].
- Reflecting [tex]\( -\sqrt{x} \)[/tex] across the y-axis changes [tex]\( x \)[/tex] to [tex]\( -x \)[/tex], resulting in [tex]\( -\sqrt{-x} \)[/tex].
- However, [tex]\( -\sqrt{-x} \)[/tex] is not a real number for [tex]\( x \geq 0 \)[/tex], so we need to properly assess [tex]\( g(x) \)[/tex].
Considering the correct process, [tex]\( g(x) = -\sqrt{x} \)[/tex] provides a reasonable reflection relationship. Thus, [tex]\( g(x) = -\sqrt{x} \)[/tex].
2. Domains and Ranges of the Functions:
- For [tex]\( f(x) = \sqrt{x} \)[/tex]:
- Domain: [tex]\( x \geq 0 \)[/tex]
- Range: [tex]\( y \geq 0 \)[/tex]
- For [tex]\( g(x) = -\sqrt{x} \)[/tex]:
- Domain: [tex]\( x \geq 0 \)[/tex] (same as [tex]\( f(x) \)[/tex])
- Range: [tex]\( y \leq 0 \)[/tex]
3. Validating the Given Statements:
- The functions have the same range.
- This statement is False. The range of [tex]\( f(x) \)[/tex] is [tex]\( y \geq 0 \)[/tex] and the range of [tex]\( g(x) \)[/tex] is [tex]\( y \leq 0 \)[/tex]. They are not the same.
- The functions have the same domains.
- This statement is True. Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the domain [tex]\( x \geq 0 \)[/tex].
- The only value that is in the domains of both functions is 0.
- This statement is False. Since both functions have the same domain of [tex]\( x \geq 0 \)[/tex], all values [tex]\( x \geq 0 \)[/tex] are in the domains of both functions, not only 0.
- There are no values that are in the ranges of both functions.
- This statement is True. The range of [tex]\( f(x) \)[/tex] is [tex]\( y \geq 0 \)[/tex] and the range of [tex]\( g(x) \)[/tex] is [tex]\( y \leq 0 \)[/tex]. No value can be simultaneously non-negative and non-positive, so there are no overlapping values in the ranges.
- The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0.
- This statement is True. The domain of [tex]\( g(x) \)[/tex] is [tex]\( x \geq 0 \)[/tex].
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
- This statement is True. The range of [tex]\( g(x) \)[/tex] is [tex]\( y \leq 0 \)[/tex].
Therefore, the correct and true statements about the functions are:
- The functions have the same domains.
- There are no values that are in the ranges of both functions.
- The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0.
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
