Answer :
To determine which graph represents the function [tex]\( f(x) \)[/tex], let's break down the piecewise function and analyze its characteristics step-by-step:
### Step 1: Analyze the function for [tex]\( 0 \leq x < 3 \)[/tex]
For this range, the function is given by:
[tex]\[ f(x) = 3 \sqrt{x + 1} \][/tex]
- Domain: [tex]\(0 \leq x < 3\)[/tex]
- Range: Determine the minimum value and maximum value for this part.
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 \sqrt{0 + 1} = 3 \sqrt{1} = 3 \][/tex]
- As [tex]\( x \)[/tex] approaches 3 (but is less than 3):
[tex]\[ f(x) = 3 \sqrt{3 + 1} = 3 \sqrt{4} = 3 \times 2 = 6 \][/tex]
- Thus, for [tex]\( 0 \leq x < 3 \)[/tex], the range is [tex]\([3, 6)\)[/tex]
### Step 2: Analyze the function for [tex]\( 3 \leq x \leq 5 \)[/tex]
For this range, the function is given by:
[tex]\[ f(x) = 5 - x \][/tex]
- Domain: [tex]\(3 \leq x \leq 5\)[/tex]
- Range: Determine the minimum value and maximum value for this part.
- At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 5 - 3 = 2 \][/tex]
- At [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 5 - 5 = 0 \][/tex]
- Thus, for [tex]\( 3 \leq x \leq 5 \)[/tex], the range is [tex]\([0, 2]\)[/tex]
### Step 3: Combine the pieces
The full function [tex]\( f(x) \)[/tex] on the domain [tex]\(0 \leq x \leq 5\)[/tex] encompasses:
- [tex]\(3 \leq f(x) < 6\)[/tex] for [tex]\(0 \leq x < 3\)[/tex]
- [tex]\(0 \leq f(x) \leq 2\)[/tex] for [tex]\(3 \leq x \leq 5\)[/tex]
### Step 4: Check continuity and values at critical points
- At [tex]\( x = 3 \)[/tex], switch between the two pieces:
[tex]\[ \text{For } x \to 3^-: \ f(3^-) = 3 \sqrt{3 + 1} = 6 \][/tex]
[tex]\[ \text{For } x = 3: \ f(3) = 5 - 3 = 2 \][/tex]
There is a discontinuity at [tex]\( x = 3 \)[/tex].
### Visualization:
1. From [tex]\( x = 0 \)[/tex] to [tex]\( x < 3 \)[/tex], the graph should show a curve starting at point (0, 3) and approaching the point (3, 6).
2. From [tex]\( x = 3 \)[/tex] to [tex]\( x = 5 \)[/tex], the graph should have a downward line segment starting at point (3, 2) and ending at point (5, 0).
By comparing the characteristics above to the given options for graphs:
- Graph A should show a curve from (0, 3) to near (3, 6) and a sharp drop to (3, 2), then a linear decline to (5, 0).
- Graph B, C, and D should be checked similarly.
So:
Based on the piecewise function [tex]\( f(x) \)[/tex], the correct graph will match these conditions.
Option [tex]\( \boxed{A} \)[/tex] is the correct graph of the given function. This option shows the correct behavior of the function as a combination of a square root function followed by a linear decrease after [tex]\( x = 3 \)[/tex].
### Step 1: Analyze the function for [tex]\( 0 \leq x < 3 \)[/tex]
For this range, the function is given by:
[tex]\[ f(x) = 3 \sqrt{x + 1} \][/tex]
- Domain: [tex]\(0 \leq x < 3\)[/tex]
- Range: Determine the minimum value and maximum value for this part.
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 \sqrt{0 + 1} = 3 \sqrt{1} = 3 \][/tex]
- As [tex]\( x \)[/tex] approaches 3 (but is less than 3):
[tex]\[ f(x) = 3 \sqrt{3 + 1} = 3 \sqrt{4} = 3 \times 2 = 6 \][/tex]
- Thus, for [tex]\( 0 \leq x < 3 \)[/tex], the range is [tex]\([3, 6)\)[/tex]
### Step 2: Analyze the function for [tex]\( 3 \leq x \leq 5 \)[/tex]
For this range, the function is given by:
[tex]\[ f(x) = 5 - x \][/tex]
- Domain: [tex]\(3 \leq x \leq 5\)[/tex]
- Range: Determine the minimum value and maximum value for this part.
- At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 5 - 3 = 2 \][/tex]
- At [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 5 - 5 = 0 \][/tex]
- Thus, for [tex]\( 3 \leq x \leq 5 \)[/tex], the range is [tex]\([0, 2]\)[/tex]
### Step 3: Combine the pieces
The full function [tex]\( f(x) \)[/tex] on the domain [tex]\(0 \leq x \leq 5\)[/tex] encompasses:
- [tex]\(3 \leq f(x) < 6\)[/tex] for [tex]\(0 \leq x < 3\)[/tex]
- [tex]\(0 \leq f(x) \leq 2\)[/tex] for [tex]\(3 \leq x \leq 5\)[/tex]
### Step 4: Check continuity and values at critical points
- At [tex]\( x = 3 \)[/tex], switch between the two pieces:
[tex]\[ \text{For } x \to 3^-: \ f(3^-) = 3 \sqrt{3 + 1} = 6 \][/tex]
[tex]\[ \text{For } x = 3: \ f(3) = 5 - 3 = 2 \][/tex]
There is a discontinuity at [tex]\( x = 3 \)[/tex].
### Visualization:
1. From [tex]\( x = 0 \)[/tex] to [tex]\( x < 3 \)[/tex], the graph should show a curve starting at point (0, 3) and approaching the point (3, 6).
2. From [tex]\( x = 3 \)[/tex] to [tex]\( x = 5 \)[/tex], the graph should have a downward line segment starting at point (3, 2) and ending at point (5, 0).
By comparing the characteristics above to the given options for graphs:
- Graph A should show a curve from (0, 3) to near (3, 6) and a sharp drop to (3, 2), then a linear decline to (5, 0).
- Graph B, C, and D should be checked similarly.
So:
Based on the piecewise function [tex]\( f(x) \)[/tex], the correct graph will match these conditions.
Option [tex]\( \boxed{A} \)[/tex] is the correct graph of the given function. This option shows the correct behavior of the function as a combination of a square root function followed by a linear decrease after [tex]\( x = 3 \)[/tex].
