Use the image to answer the question.

Question

One solid and one dotted curve are drawn in a coordinate plane. The axes range from negative 2 to 2, both by 1-unit increments.

Compare the graphs of f(x)=−x−−√ and g(x)=−x−−√3. Which of the following features about the graphs is true?

(1 point)
Responses

The graphs have the same ranges.
The graphs have the same ranges.

The graphs are both decreasing.
The graphs are both decreasing.

The graphs have the same domains.
The graphs have the same domains.

The graphs are both increasing.
The graphs are both increasing.
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Answer 1

To compare the functions \( f(x) = -\sqrt{x} \) and \( g(x) = -\sqrt{3} \):

1. Domains:

   • The function \( f(x) = -\sqrt{x} \) is defined for \( x \geq 0 \) (since you cannot take the square root of a negative number). Therefore, the domain of \( f(x) \) is \( [0, 2) \).
   • The function \( g(x) = -\sqrt{3} \) is a constant function, meaning it is defined for all values of \( x \). Thus, the domain of \( g(x) \) is \( (-\infty, \infty) \).

   Conclusion: The graphs do not have the same domain.

2. Ranges:

   • The range of \( f(x) = -\sqrt{x} \) starts at 0 when \( x = 0 \) and goes to negative infinity as \( x \) increases. So, the range is \( (-\infty, 0] \).
   • For \( g(x) = -\sqrt{3} \), the output is the constant value \( -\sqrt{3} \), so the range is simply \( {-\sqrt{3}} \).

   Conclusion: The graphs do not have the same range.

3. Increasing/Decreasing:

   • The function \( f(x) = -\sqrt{x} \) is decreasing because as \( x \) increases, \( f(x) \) decreases (i.e., becomes more negative).
   • The function \( g(x) = -\sqrt{3} \) is constant, which means it is neither increasing nor decreasing.

   Conclusion: The graphs are not both decreasing.

Based on this analysis, the only correct feature of the graphs that can be concluded is:

None of the provided statements are true.