Answer :
To find which graph represents the function [tex]\( S(x) = \frac{1}{2} \sqrt{x + 3} \)[/tex], we can analyze the function [tex]\( S(x) \)[/tex] and its behavior.
### Step-by-Step Solution
1. Understanding the function [tex]\( S(x) = \frac{1}{2} \sqrt{x + 3} \)[/tex]:
- This function describes how the side length of the smaller sticker [tex]\( S(x) \)[/tex] changes with respect to the area of the larger sticker [tex]\( x \)[/tex].
2. Domain of the function:
- Since we are dealing with real-world measurements of area, [tex]\( x \)[/tex] should be non-negative. Hence, [tex]\( x \geq 0 \)[/tex].
3. Behavior of the function:
- Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ S(0) = \frac{1}{2} \sqrt{0 + 3} = \frac{1}{2} \sqrt{3} \approx 0.866 \][/tex]
- Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[ S(1) = \frac{1}{2} \sqrt{1 + 3} = \frac{1}{2} \sqrt{4} = \frac{1}{2} \cdot 2 = 1 \][/tex]
- Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[ S(4) = \frac{1}{2} \sqrt{4 + 3} = \frac{1}{2} \sqrt{7} \approx 1.322 \][/tex]
- Substitute [tex]\( x = 9 \)[/tex]:
[tex]\[ S(9) = \frac{1}{2} \sqrt{9 + 3} = \frac{1}{2} \sqrt{12} = \frac{1}{2} \cdot 2 \sqrt{3} \approx 1.732 \][/tex]
4. Behavior at large values of [tex]\( x \)[/tex]:
- As [tex]\( x \)[/tex] becomes very large, [tex]\( S(x) \)[/tex] also increases, but at a decreasing rate due to the square root function.
- For instance, at [tex]\( x = 100 \)[/tex]:
[tex]\[ S(100) = \frac{1}{2} \sqrt{100 + 3} = \frac{1}{2} \sqrt{103} \approx \frac{1}{2} \cdot 10.148 = 5.074 \][/tex]
5. Generation and interpretation of data points:
- We generate several points of [tex]\( S(x) \)[/tex] over the range from [tex]\( x = 0 \)[/tex] to [tex]\( x = 100 \)[/tex].
- Some key points observed would be:
[tex]\[ S(0) \approx 0.866, \quad S(10) \approx 1.150, \quad S(20) \approx 1.366, \quad S(50) \approx 1.908, \quad S(100) \approx 2.525 \][/tex]
- Notice the smooth and continuous increase of the side length [tex]\( S(x) \)[/tex] with increasing [tex]\( x \)[/tex].
6. Identifying the correct graph:
- Based on the above evaluations and key points, the correct graph should show a curve that starts at approximately [tex]\( 0.866 \)[/tex] when [tex]\( x = 0 \)[/tex], gradually increases, and continues to rise, but at a decelerating rate due to the nature of the square root function.
Comparing these observations with the provided options, the correct graph will match the behavior outlined:
- Starts around [tex]\( 0.866 \)[/tex] when [tex]\( x = 0 \)[/tex].
- Gradually increases and shows a smooth, continuous curve.
- Clearly represents the nature of [tex]\( \frac{1}{2} \sqrt{x + 3} \)[/tex].
Thus, the graph that matches these characteristics is the correct one representing the function [tex]\( S(x) \)[/tex].
### Step-by-Step Solution
1. Understanding the function [tex]\( S(x) = \frac{1}{2} \sqrt{x + 3} \)[/tex]:
- This function describes how the side length of the smaller sticker [tex]\( S(x) \)[/tex] changes with respect to the area of the larger sticker [tex]\( x \)[/tex].
2. Domain of the function:
- Since we are dealing with real-world measurements of area, [tex]\( x \)[/tex] should be non-negative. Hence, [tex]\( x \geq 0 \)[/tex].
3. Behavior of the function:
- Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ S(0) = \frac{1}{2} \sqrt{0 + 3} = \frac{1}{2} \sqrt{3} \approx 0.866 \][/tex]
- Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[ S(1) = \frac{1}{2} \sqrt{1 + 3} = \frac{1}{2} \sqrt{4} = \frac{1}{2} \cdot 2 = 1 \][/tex]
- Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[ S(4) = \frac{1}{2} \sqrt{4 + 3} = \frac{1}{2} \sqrt{7} \approx 1.322 \][/tex]
- Substitute [tex]\( x = 9 \)[/tex]:
[tex]\[ S(9) = \frac{1}{2} \sqrt{9 + 3} = \frac{1}{2} \sqrt{12} = \frac{1}{2} \cdot 2 \sqrt{3} \approx 1.732 \][/tex]
4. Behavior at large values of [tex]\( x \)[/tex]:
- As [tex]\( x \)[/tex] becomes very large, [tex]\( S(x) \)[/tex] also increases, but at a decreasing rate due to the square root function.
- For instance, at [tex]\( x = 100 \)[/tex]:
[tex]\[ S(100) = \frac{1}{2} \sqrt{100 + 3} = \frac{1}{2} \sqrt{103} \approx \frac{1}{2} \cdot 10.148 = 5.074 \][/tex]
5. Generation and interpretation of data points:
- We generate several points of [tex]\( S(x) \)[/tex] over the range from [tex]\( x = 0 \)[/tex] to [tex]\( x = 100 \)[/tex].
- Some key points observed would be:
[tex]\[ S(0) \approx 0.866, \quad S(10) \approx 1.150, \quad S(20) \approx 1.366, \quad S(50) \approx 1.908, \quad S(100) \approx 2.525 \][/tex]
- Notice the smooth and continuous increase of the side length [tex]\( S(x) \)[/tex] with increasing [tex]\( x \)[/tex].
6. Identifying the correct graph:
- Based on the above evaluations and key points, the correct graph should show a curve that starts at approximately [tex]\( 0.866 \)[/tex] when [tex]\( x = 0 \)[/tex], gradually increases, and continues to rise, but at a decelerating rate due to the nature of the square root function.
Comparing these observations with the provided options, the correct graph will match the behavior outlined:
- Starts around [tex]\( 0.866 \)[/tex] when [tex]\( x = 0 \)[/tex].
- Gradually increases and shows a smooth, continuous curve.
- Clearly represents the nature of [tex]\( \frac{1}{2} \sqrt{x + 3} \)[/tex].
Thus, the graph that matches these characteristics is the correct one representing the function [tex]\( S(x) \)[/tex].
