Question

Graph the square root functions f(x)=x−2−−−−√ and g(x)=2−x−−−−√
. Which function has a y-intercept? (1 point)
Responses

f(x)
f(x)

g(x)
g(x)

neither f(x) or g(x)
neither f(x) or g(x)

both f(x) and g(x)

Answer 1

To analyze the functions \( f(x) = \sqrt{x - 2} \) and \( g(x) = \sqrt{2 - x} \), we need to determine their domains and y-intercepts.

### 1. Analyzing \( f(x) = \sqrt{x - 2} \)
- **Domain:** The expression \( x - 2 \) must be non-negative, so:
\[
x - 2 \geq 0 \Rightarrow x \geq 2
\]
Thus, the domain of \( f(x) \) is \( [2, \infty) \).

- **Y-intercept:** To find the y-intercept, we evaluate \( f(0) \):
\[
f(0) = \sqrt{0 - 2} \quad \text{(not defined, since } \sqrt{-2} \text{ is not real)}
\]
Therefore, \( f(x) \) has no y-intercept.

### 2. Analyzing \( g(x) = \sqrt{2 - x} \)
- **Domain:** The expression \( 2 - x \) must be non-negative, so:
\[
2 - x \geq 0 \Rightarrow x \leq 2
\]
Thus, the domain of \( g(x) \) is \( (-\infty, 2] \).

- **Y-intercept:** To find the y-intercept, we evaluate \( g(0) \):
\[
g(0) = \sqrt{2 - 0} = \sqrt{2}
\]
Therefore, \( g(x) \) has a y-intercept at \( (0, \sqrt{2}) \).

### Conclusion:
- **Function with a y-intercept:** Only \( g(x) \) has a y-intercept.

Thus, the answer is **g(x)**.