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 domains.
The graphs have the same domains.

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

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

The graphs are both increasing.
The graphs are both increasing.

Answer 1

Let's analyze the two functions:

1. \( f(x) = -\sqrt{-x} \)
2. \( g(x) = -\sqrt{-x} + 3 \)

### Domains:

- **For \( f(x) \)**:
- The expression \(-x\) must be non-negative for the square root to be defined, which means \( -x \geq 0 \) or \( x \leq 0 \).
- Thus, the domain of \( f(x) \) is \( (-\infty, 0] \).

- **For \( g(x) \)**:
- The same condition applies: \( -x \geq 0 \) requires \( x \leq 0 \).
- Therefore, the domain of \( g(x) \) is also \( (-\infty, 0] \).

### Ranges:

- **Range of \( f(x) \)**:
- As \( x \) approaches \( 0 \) from the left, \( f(x) \) approaches \( 0 \), and as \( x \) goes to negative infinity, \( f(x) \) goes to positive infinity.
- Thus, the range of \( f(x) \) is \( (-\infty, 0] \).

- **Range of \( g(x) \)**:
- For \( g(x) = -\sqrt{-x} + 3 \), as \( f(x) \) approaches \( 0 \) (when \( x \) approaches \( 0 \)), \( g(x) \) approaches \( 3 \).
- As \( x \) goes to negative infinity, \( g(x) \) approaches \( 3 \) from below, resulting in the range of \( g(x) \) being \( (-\infty, 3] \).

### Monotonicity (Increasing/Decreasing):

- **For both functions**:
- The square root function is decreasing, and since both functions have a negative component, they will also be decreasing overall as \( x \) decreases.
- Therefore, both \( f(x) \) and \( g(x) \) are decreasing functions.

### Overall Results:

1. **The graphs have the same domains.** (True: both are \( (-\infty, 0] \))
2. **The graphs have the same ranges.** (False: \( f(x) \) is \( (-\infty, 0] \) and \( g(x) \) is \( (-\infty, 3] \))
3. **The graphs are both decreasing.** (True: both are decreasing)
4. **The graphs are both increasing.** (False: they are both decreasing)

### Conclusion:
The true statements are:
- The graphs have the same domains.
- The graphs are both decreasing.

Since the question asked for "which of the following features about the graphs is true," the correct responses would be that they have the same domains and that both graphs are decreasing.