Graph the function y=x−3−−−−√+1

Question

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:

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