Sure, let's go through the steps to graph the function \( f(x) = 2 \sqrt{x+3} \) in detail.
### Step-by-Step Solution:
1. Identify the Domain:
- Since we have a square root function, the expression inside the square root must be non-negative.
- Therefore, \( x + 3 \geq 0 \).
- Solving this inequality, we get \( x \geq -3 \).
- So, the domain of \( f(x) \) is \( x \in [-3, \infty) \).
2. Identify Key Points:
- It can be useful to identify some specific points on the function to help with the plot.
- Let's calculate \( f(x) \) for a few values of \( x \):
[tex]\[
\begin{aligned}
&f(-3) = 2 \sqrt{-3 + 3} = 2 \sqrt{0} = 0, \\
&f(0) = 2 \sqrt{0 + 3} = 2 \sqrt{3} \approx 3.46, \\
&f(1) = 2 \sqrt{1 + 3} = 2 \sqrt{4} = 4, \\
&f(4) = 2 \sqrt{4 + 3} = 2 \sqrt{7} \approx 5.29.
\end{aligned}
\][/tex]
3. Sketch the Graph:
- Now we plot the function using the domain and key points identified above.
- The function starts at \( x = -3 \) with \( f(-3) = 0 \) and increases as \( x \) increases.
- The function is not defined for \( x < -3 \).
4. Observing Function Behavior:
- For very large values of \( x \), \( f(x) \) will also increase but at a diminishing rate because the square root function grows slower than a linear function.
- As \( x \) approaches \(-3 \) from the right, \( f(x) \) approaches 0.
Here's a simple sketch to visualize it:
1. Draw the x-axis and y-axis.
2. Mark the domain starting from \( x = -3 \).
3. Plot the key points:
- \( (-3, 0) \),
- \( (0, \approx3.46) \),
- \( (1, 4) \),
- \( (4, \approx5.29) \).
4. Draw a smooth curve through these points, noting that it starts at the origin of the domain (\( x = -3 \)) and gradually curves upwards as \( x \) increases.
By following these steps, you should have a clear understanding and a precise graph of the function [tex]\( f(x) = 2 \sqrt{x+3} \)[/tex].
