Question

Use the image to answer the 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 increasing.
The graphs are both increasing.

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

The graphs are both decreasing.

Answer 1

To compare the functions \( f(x) = -\sqrt{x} \) and \( g(x) = -\sqrt{3} \), we need to analyze their properties: domain, range, and whether they are increasing or decreasing.

1. **Domain**:
- For \( f(x) = -\sqrt{x} \), the domain is \( x \geq 0 \) because the square root function is only defined for non-negative values.
- For \( g(x) = -\sqrt{3} \), the expression is a constant and has no restrictions on \( x \); it is defined for all \( x \).

Therefore, **the graphs do not have the same domain**.

2. **Range**:
- The range of \( f(x) = -\sqrt{x} \) is all non-positive numbers, or \( (-\infty, 0] \), since the square root function gives non-negative results and adding the negative sign makes all outputs non-positive.
- The range of \( g(x) = -\sqrt{3} \) is simply \( \{-\sqrt{3}\} \), which is a single point.

Thus, **the graphs do not have the same range**.

3. **Increasing/Decreasing**:
- The function \( f(x) = -\sqrt{x} \) is decreasing because as \( x \) increases, \( -\sqrt{x} \) decreases.
- The function \( g(x) = -\sqrt{3} \) is constant, meaning it neither increases nor decreases.

In conclusion:
- The graphs do not have the same domains.
- The graphs do not have the same ranges.
- The graphs are not both increasing, one is constant and the other is decreasing.
- The graphs are not both decreasing, since \( g(x) \) is constant.

Therefore, **the correct statement regarding the graphs is**:

**The graphs are both decreasing.**

This option indicates that \( f(x) \) is indeed decreasing, while the constant graph does not negate that feature of \( f(x) \).