Question

Question
Use the image to answer the question.

One solid and one dotted curve are drawn in a coordinate plane. The x-axis ranges from negative 3 to 3 and the y-axis ranges from negative 4 to 2, both by 1-unit increments.

Compare the graphs of f(x)=x+1−−−−√−2
and g(x)=x+1−−−−√3−2
. Identify which of the following features is true about the graphs.

(1 point)
Responses

The graphs have the same x-intercept.
The graphs have the same x -intercept.

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

The graphs have the same y-intercept.
The graphs have the same y -intercept.

The graphs have the same ranges.
The graphs have the same ranges.
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Answer 1

To compare the functions \( f(x) = \sqrt{x + 1} - 2 \) and \( g(x) = \sqrt{x + 1} + 3 - 2 \), we can analyze their features: x-intercepts, domains, y-intercepts, and ranges.

1. **X-intercepts**:
- To find the x-intercepts, set \( f(x) = 0 \) and \( g(x) = 0 \).
- For \( f(x) = \sqrt{x + 1} - 2 = 0 \):
\[
\sqrt{x + 1} = 2 \implies x + 1 = 4 \implies x = 3.
\]
- For \( g(x) = \sqrt{x + 1} + 1 = 0 \):
\[
\sqrt{x + 1} = -1 \text{ (not possible, since the square root cannot be negative)}.
\]
- Thus, the x-intercept of \( g(x) \) does not exist and is different from that of \( f(x) \).

2. **Domains**:
- Both functions contain \( \sqrt{x + 1} \), which requires \( x + 1 \geq 0 \) or \( x \geq -1 \).
- So the domain for both functions is \( x \geq -1 \).

3. **Y-intercepts**:
- To find the y-intercepts, evaluate \( f(0) \) and \( g(0) \):
- For \( f(0) \):
\[
f(0) = \sqrt{0 + 1} - 2 = 1 - 2 = -1.
\]
- For \( g(0) \):
\[
g(0) = \sqrt{0 + 1} + 3 - 2 = 1 + 1 = 2.
\]
- Thus, the y-intercepts are \( (-1) \) for \( f \) and \( (2) \) for \( g \), so they are different.

4. **Ranges**:
- The range of \( f(x) \):
\[
f(x) \text{ starts at } -1 \text{ and increases, approaching infinity. Thus, its range is } [-1, \infty).
\]
- The range of \( g(x) \):
\[
g(x) \text{ starts at } 2 \text{ and also increases, approaching infinity. Thus, its range is } [2, \infty).
\]
- Since the lowest values in their ranges are different, this means their ranges are also different.

### Conclusion:
The only feature that is true about the graphs is that they have the **same domains**. Therefore, the correct answer is:

**The graphs have the same domains.**