To represent the graph of the function [tex]\( f(x) = \sqrt{x + 2} - 3 \)[/tex], let's break down the steps to understand its key features and plot points:
### Step-by-Step Solution:
1. Function Definition:
The function we are dealing with is:
[tex]\[
f(x) = \sqrt{x + 2} - 3
\][/tex]
2. Domain:
For the square root to be defined, the argument inside must be greater than or equal to zero:
[tex]\[
x + 2 \geq 0 \implies x \geq -2
\][/tex]
Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq -2 \)[/tex].
3. Range:
The smallest value of [tex]\( \sqrt{x + 2} \)[/tex] is 0 when [tex]\( x = -2 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x + 2} \)[/tex] increases without bound. So:
[tex]\[
\sqrt{x + 2} - 3 \geq -3
\][/tex]
The range of [tex]\( f(x) \)[/tex] is [tex]\( y \geq -3 \)[/tex].
4. Intercepts:
- Y-intercept: To find the y-intercept, we set [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = \sqrt{0 + 2} - 3 = \sqrt{2} - 3 \approx -1.586
\][/tex]
- X-intercept: To find the x-intercept, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
0 = \sqrt{x + 2} - 3 \implies \sqrt{x + 2} = 3 \implies x + 2 = 9 \implies x = 7
\][/tex]
5. Behavior at Key Points:
- [tex]\( f(-2) = \sqrt{-2 + 2} - 3 = 0 - 3 = -3 \)[/tex]
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( f(x) \)[/tex] increases because √(x+2) grows indefinitely.
Let's now visualize the values plotted:
- For [tex]\( x \)[/tex] values from [tex]\(-2\)[/tex] to [tex]\(10\)[/tex], corresponding [tex]\( y \)[/tex] values approximately are:
[tex]\[
\begin{aligned}
f(x = -2) & = -3, \\
f(x = 0) & \approx -1.586, \\
f(x = 1) & \approx -1.000, \\
f(x = 2) & \approx -0.586, \\
f(x = 7) & = 0.
\end{aligned}
\][/tex]
6. Graph Shape:
- The graph starts at (-2, -3).
- It moves upwards, passing through points such as (0, -1.586), (1, -1), (2, -0.586), and hits zero at (7, 0).
- From there, it continues to increase without bound as [tex]\( x \)[/tex] increases.
Here’s a visual representation of [tex]\( f(x) = \sqrt{x + 2} - 3 \)[/tex]:
- The graph originates from (-2, -3).
- It has a smooth, continuous curve rising towards the right.
- The curve gradually increases towards infinity, starting steep due to the square root nature, and passes through key points calculated, ensuring both domain and range is respected.
Thus, when looking for a graph of [tex]\( f(x) = \sqrt{x + 2} - 3 \)[/tex], you should see a curve that meets these characteristics: originates at (-2, -3), passes through (7, 0), and rises upwards towards infinity.
