To compare the graphs of the functions \( f(x) = -\sqrt{x} \) and \( g(x) = -\sqrt{3} \), let's analyze each function:
-
Domain:
- The function \( f(x) = -\sqrt{x} \) is defined for \( x \geq 0 \) (because the square root function requires non-negative inputs). Thus, its domain is \( [0, \infty) \).
- The function \( g(x) = -\sqrt{3} \) is a constant function, meaning it is defined for all \( x \) (i.e., \( (-\infty, \infty) \)).
Since their domains are different, we can eliminate the option "The graphs have the same domains."
-
Range:
- The range of \( f(x) = -\sqrt{x} \) is \( (-\infty, 0] \) because the square root function produces non-negative values and the negative sign flips them to be non-positive.
- The range of \( g(x) = -\sqrt{3} \) is just the single value \( -\sqrt{3} \), so the range is \( {-\sqrt{3}} \).
Since the ranges are also different, we can eliminate the option "The graphs have the same ranges."
-
Increasing or Decreasing:
- The function \( f(x) = -\sqrt{x} \) is decreasing because as \( x \) increases, \( -\sqrt{x} \) moves downwards.
- The function \( g(x) = -\sqrt{3} \) is constant (neither increasing nor decreasing).
Therefore, we can eliminate the option "The graphs are both increasing" and "The graphs are both decreasing."
Based on this analysis, the only correct option that can be drawn is that:
None of the statements regarding the comparisons of these two functions are true.
However, if strictly adhering to the original choices, the most fitting response considering the specific details provided would be:
The graphs have the same domains. (This is overruled since they do not share any domains)
To restate, under proper evaluation, "The graphs have different domains, ranges, and increasing behavior."
None of the provided choices should be regarded as true since the domains and ranges are different, and neither of these functions are both increasing or both decreasing.