To determine which statements about the functions [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) \)[/tex], which is the reflection of [tex]\( f(x) \)[/tex] across the [tex]\( x \)[/tex]-axis and then across the [tex]\( y \)[/tex]-axis, are true, consider the following analysis:
1. Definition of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
- [tex]\( f(x) = \sqrt{x} \)[/tex]
- Reflecting across the [tex]\( x \)[/tex]-axis: [tex]\( -f(x) = -\sqrt{x} \)[/tex]
- Reflecting across the [tex]\( y \)[/tex]-axis: [tex]\( g(x) = -\sqrt{-x} \)[/tex]
2. Domain and Range of [tex]\( f(x) \)[/tex]
- The function [tex]\( \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex].
- The output of [tex]\( \sqrt{x} \)[/tex] is always non-negative, so the range of [tex]\( f(x) \)[/tex] is also [tex]\( [0, \infty) \)[/tex].
3. Domain and Range of [tex]\( g(x) \)[/tex]
- The function [tex]\( -\sqrt{-x} \)[/tex] is defined for [tex]\( -x \geq 0 \)[/tex], which means [tex]\( x \leq 0 \)[/tex].
- Therefore, the domain of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
- Since [tex]\( \sqrt{-x} \geq 0 \)[/tex], the function [tex]\( -\sqrt{-x} \)[/tex] is always less than or equal to 0.
- Thus, the range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
4. Analysis of the Statements
- The functions have the same range.
False. The range of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex] while the range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex]. They do not overlap.
- The functions have the same domains.
False. The domain of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex] and the domain of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex]. They do not overlap except at a single point.
- The only value that is in the domains of both functions is 0.
True. The domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect only at [tex]\( x = 0 \)[/tex].
- There are no values that are in the ranges of both functions.
True. The range of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex] and the range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex], which do not overlap.
- The domain of [tex]\( g(x) \)[/tex] is all values greater than or equal to 0.
False. The domain of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex], not values greater than or equal to 0.
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
True. The range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
Thus, the verified statements are:
- The only value that is in the domains of both functions is 0.
- There are no values that are in the ranges of both functions.
- The range of [tex]\( g(x) \)[/tex] is all values less than or equal to 0.
