Answer :
To analyze the given problem, let's construct the functions step by step and evaluate each statement regarding [tex]\( f(x) = \sqrt{x} \)[/tex] and its reflection [tex]\( g(x) \)[/tex].
1. Reflecting [tex]\( f(x) = \sqrt{x} \)[/tex] across the x-axis:
Reflecting a function across the x-axis involves changing the sign of the function's output:
[tex]\[ h(x) = -\sqrt{x} \][/tex]
2. Reflecting [tex]\( h(x) = -\sqrt{x} \)[/tex] across the y-axis:
Reflecting a function across the y-axis involves changing the sign of the function's input:
[tex]\[ g(x) = h(-x) = -\sqrt{-x} \][/tex]
However, this function as given is defined for [tex]\( x \leq 0 \)[/tex] because the square root function is only defined for non-negative inputs. Thus:
[tex]\[ g(x) = -\sqrt{-x} \][/tex]
With [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] defined, we can check each statement one by one.
### Statement Analysis:
1. The functions have the same range.
- The range of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex].
- The range of [tex]\( g(x) = -\sqrt{-x} \)[/tex] is [tex]\((-\infty, 0]\)[/tex].
These ranges are different, so this statement is false.
2. The functions have the same domains.
- The domain of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex].
- The domain of [tex]\( g(x) = -\sqrt{-x} \)[/tex] is [tex]\([0, \infty)\)[/tex].
These domains are the same, so this statement is true.
3. The only value that is in the domains of both functions is 0.
- As mentioned, both functions are defined for [tex]\( x \geq 0 \)[/tex].
Thus, this statement is false. They share the entire interval [tex]\([0,\infty)\)[/tex], not just the point 0.
4. There are no values that are in the ranges of both functions.
- The range of [tex]\( f(x) \)[/tex] is [tex]\([0, \infty)\)[/tex].
- The range of [tex]\( g(x) \)[/tex] is [tex]\((-\infty, 0]\)[/tex].
There are no values common between these ranges, so this statement is true.
5. The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0.
- The domain of [tex]\( g(x) \)[/tex] is indeed [tex]\([0, \infty)\)[/tex].
So, this statement is true.
6. The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
- The range of [tex]\( g(x) \)[/tex] is indeed [tex]\((-\infty, 0]\)[/tex].
So, this statement is true.
### Conclusion
Based on the detailed analysis, the correct answers are:
- The functions have the same range. [tex]\(\quad (False)\)[/tex]
- The functions have the same domains. [tex]\(\quad (True)\)[/tex]
- The only value that is in the domains of both functions is 0. [tex]\(\quad (False)\)[/tex]
- There are no values that are in the ranges of both functions. [tex]\(\quad (True)\)[/tex]
- The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0. [tex]\(\quad (True)\)[/tex]
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0. [tex]\(\quad (True)\)[/tex]
Thus, the solution is:
[tex]\[ (0, 1, 0, 1, 1, 1) \][/tex]
1. Reflecting [tex]\( f(x) = \sqrt{x} \)[/tex] across the x-axis:
Reflecting a function across the x-axis involves changing the sign of the function's output:
[tex]\[ h(x) = -\sqrt{x} \][/tex]
2. Reflecting [tex]\( h(x) = -\sqrt{x} \)[/tex] across the y-axis:
Reflecting a function across the y-axis involves changing the sign of the function's input:
[tex]\[ g(x) = h(-x) = -\sqrt{-x} \][/tex]
However, this function as given is defined for [tex]\( x \leq 0 \)[/tex] because the square root function is only defined for non-negative inputs. Thus:
[tex]\[ g(x) = -\sqrt{-x} \][/tex]
With [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] defined, we can check each statement one by one.
### Statement Analysis:
1. The functions have the same range.
- The range of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex].
- The range of [tex]\( g(x) = -\sqrt{-x} \)[/tex] is [tex]\((-\infty, 0]\)[/tex].
These ranges are different, so this statement is false.
2. The functions have the same domains.
- The domain of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex].
- The domain of [tex]\( g(x) = -\sqrt{-x} \)[/tex] is [tex]\([0, \infty)\)[/tex].
These domains are the same, so this statement is true.
3. The only value that is in the domains of both functions is 0.
- As mentioned, both functions are defined for [tex]\( x \geq 0 \)[/tex].
Thus, this statement is false. They share the entire interval [tex]\([0,\infty)\)[/tex], not just the point 0.
4. There are no values that are in the ranges of both functions.
- The range of [tex]\( f(x) \)[/tex] is [tex]\([0, \infty)\)[/tex].
- The range of [tex]\( g(x) \)[/tex] is [tex]\((-\infty, 0]\)[/tex].
There are no values common between these ranges, so this statement is true.
5. The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0.
- The domain of [tex]\( g(x) \)[/tex] is indeed [tex]\([0, \infty)\)[/tex].
So, this statement is true.
6. The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
- The range of [tex]\( g(x) \)[/tex] is indeed [tex]\((-\infty, 0]\)[/tex].
So, this statement is true.
### Conclusion
Based on the detailed analysis, the correct answers are:
- The functions have the same range. [tex]\(\quad (False)\)[/tex]
- The functions have the same domains. [tex]\(\quad (True)\)[/tex]
- The only value that is in the domains of both functions is 0. [tex]\(\quad (False)\)[/tex]
- There are no values that are in the ranges of both functions. [tex]\(\quad (True)\)[/tex]
- The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0. [tex]\(\quad (True)\)[/tex]
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0. [tex]\(\quad (True)\)[/tex]
Thus, the solution is:
[tex]\[ (0, 1, 0, 1, 1, 1) \][/tex]
