To determine the graph of the function [tex]\( h(x) = \sqrt{x - 2} + 3 \)[/tex], let’s analyze the key features of the function step-by-step.
1. Domain of the function:
The function involves a square root, which means the expression inside the square root, [tex]\( x - 2 \)[/tex], must be non-negative:
[tex]\[
x - 2 \geq 0 \implies x \geq 2
\][/tex]
Therefore, the domain of the function is [tex]\( [2, \infty) \)[/tex].
2. Range of the function:
Since [tex]\( \sqrt{x-2} \)[/tex] is always non-negative (as the square root of a non-negative number) and can take any value from 0 to [tex]\(\infty\)[/tex] as [tex]\( x \)[/tex] ranges from 2 to [tex]\(\infty\)[/tex], adding 3 shifts this entire range up by 3:
[tex]\[
h(x) = \sqrt{x-2} + 3 \Rightarrow \text{Range is } [3, \infty)
\][/tex]
3. Behavior at endpoints and key points:
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
h(2) = \sqrt{2 - 2} + 3 = 0 + 3 = 3
\][/tex]
So the function starts at the point [tex]\( (2, 3) \)[/tex].
- As [tex]\( x \)[/tex] increases beyond 2, [tex]\( \sqrt{x-2} \)[/tex] increases, thus [tex]\( h(x) \)[/tex] also increases.
4. Increasing nature:
For [tex]\( x > 2 \)[/tex], both [tex]\( \sqrt{x-2} \)[/tex] and [tex]\( h(x) \)[/tex] are increasing functions because the square root function is strictly increasing.
5. Graph behavior:
The graph of [tex]\( h(x) \)[/tex] will start at [tex]\( (2, 3) \)[/tex] and rise continuously without bound.
Let’s plot some additional points to clarify the graph:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
h(3) = \sqrt{3 - 2} + 3 = 1 + 3 = 4
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
h(4) = \sqrt{4 - 2} + 3 = \sqrt{2} + 3 \approx 1.41 + 3 = 4.41
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
h(6) = \sqrt{6 - 2} + 3 = 2 + 3 = 5
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
h(10) = \sqrt{10 - 2} + 3 = \sqrt{8} + 3 \approx 2.83 + 3 = 5.83
\][/tex]
From these points, we can establish the nature of the graph:
The graph of [tex]\( h(x) = \sqrt{x - 2} + 3 \)[/tex] starts at the point [tex]\( (2, 3) \)[/tex] and increases continuously, approaching higher values smoothly as [tex]\( x \)[/tex] increases. The shape resembles that of a square root function translated right by 2 units and up by 3 units.
The data given would correspond accurately to this description:
- The [tex]\( x \)[/tex]-values ranging from 2 to 10.
- The [tex]\( y \)[/tex]-values start at 3 and increase without bound, matching the points we calculated.
Thus, the graph provides a clear visualization of the function [tex]\( h(x) \)[/tex].
