Question

Graph the function y=x−3−−−−√+1
and determine its endpoint(s).(1 point)
Responses

(0,0)
left parenthesis 0 comma 0 right parenthesis

(3,−1)
left parenthesis 3 comma negative 1 right parenthesis

(3,1)
left parenthesis 3 comma 1 right parenthesis

(−3,1)

Answer 1

To graph the function \( y = \sqrt{x - 3} + 1 \), we will start by identifying its domain and range, as these will help us locate the endpoint(s).

**Step 1: Find the domain**

The expression inside the square root \( \sqrt{x - 3} \) must be non-negative:

\[
x - 3 \geq 0
\]

This gives us:

\[
x \geq 3
\]

So, the domain of the function is \( [3, \infty) \).

**Step 2: Find the range**

Since the square root function outputs non-negative values, the smallest value occurs when \( x = 3 \):

\[
y = \sqrt{3 - 3} + 1 = 0 + 1 = 1
\]

As \( x \) increases, \( \sqrt{x - 3} \) also increases, and thus \( y \) will also increase without bound. Therefore, the range is \( [1, \infty) \).

**Step 3: Determine the endpoint**

The endpoint of the graph occurs at the lower limit of the domain:

\[
(3, 1)
\]

**Conclusion:**

The endpoint of the function \( y = \sqrt{x - 3} + 1 \) is:

\[
\boxed{(3, 1)}
\]