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To graph the function [tex]\( f(x) = 3 \sqrt{x-1} \)[/tex], we will follow a step-by-step process to identify the key characteristics and plot important points on the coordinate plane.
### 1. Function Analysis
#### Domain:
The function [tex]\( f(x) = 3 \sqrt{x-1} \)[/tex] involves a square root, which means the expression inside the square root, [tex]\( x-1 \)[/tex], must be non-negative. Therefore:
[tex]\[ x - 1 \geq 0 \][/tex]
[tex]\[ x \geq 1 \][/tex]
So, the domain of the function is [tex]\( x \in [1, \infty) \)[/tex].
#### Range:
For [tex]\( x \geq 1 \)[/tex], the expression [tex]\( \sqrt{x-1} \)[/tex] can take any non-negative value from 0 to [tex]\(\infty\)[/tex]. When [tex]\( \sqrt{x-1} \)[/tex] = 0, [tex]\( f(x) = 0 \)[/tex]. As [tex]\( x \to \infty \)[/tex], [tex]\( \sqrt{x-1} \to \infty \)[/tex], making [tex]\( f(x) = 3 \sqrt{x-1} \to \infty \)[/tex]. Hence, the range of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex].
### 2. Important Points to Plot
Let's determine a few key points that lie on the function [tex]\( f(x) \)[/tex]:
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3 \sqrt{1 - 1} = 3 \times 0 = 0 \][/tex]
Point: [tex]\( (1, 0) \)[/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 3 \sqrt{2 - 1} = 3 \times 1 = 3 \][/tex]
Point: [tex]\( (2, 3) \)[/tex]
- When [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 3 \sqrt{4 - 1} = 3 \sqrt{3} \approx 3 \times 1.732 \approx 5.196 \][/tex]
Point: [tex]\( (4, 5.196) \)[/tex]
- When [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = 3 \sqrt{10 - 1} = 3 \sqrt{9} = 3 \times 3 = 9 \][/tex]
Point: [tex]\( (10, 9) \)[/tex]
### 3. Drawing the Graph
Using these points, we can sketch the graph of the function. Start by plotting the following points on the coordinate plane:
- [tex]\( (1, 0) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
- [tex]\( (4, 5.196) \)[/tex]
- [tex]\( (10, 9) \)[/tex]
These points are taken from the generated data:
[tex]\[ \begin{align*} (1.0, 0.0) & , (2.0, 3.0) \\ (4.0, 5.196) & , (10.0, 9.0) \end{align*} \][/tex]
### 4. Graph Shape Description
- The graph starts at [tex]\( (1, 0) \)[/tex] since [tex]\( f(1) = 0 \)[/tex].
- As [tex]\( x \)[/tex] increases, the function [tex]\( f(x) = 3 \sqrt{x-1} \)[/tex] grows rapidly but in a smooth, continuous curve.
- There is a gradual steepening as [tex]\( x \)[/tex] increases because the square root function increases more slowly than a linear function.
### Summary
The graph of [tex]\( f(x) = 3 \sqrt{x-1} \)[/tex] is a curve that begins at the point [tex]\( (1, 0) \)[/tex] and rises gradually. It represents a square root function scaled by a factor of 3, starting at [tex]\( x = 1 \)[/tex] and increasing continuously. The plotted points provide a guide to draw the function accurately.
### 1. Function Analysis
#### Domain:
The function [tex]\( f(x) = 3 \sqrt{x-1} \)[/tex] involves a square root, which means the expression inside the square root, [tex]\( x-1 \)[/tex], must be non-negative. Therefore:
[tex]\[ x - 1 \geq 0 \][/tex]
[tex]\[ x \geq 1 \][/tex]
So, the domain of the function is [tex]\( x \in [1, \infty) \)[/tex].
#### Range:
For [tex]\( x \geq 1 \)[/tex], the expression [tex]\( \sqrt{x-1} \)[/tex] can take any non-negative value from 0 to [tex]\(\infty\)[/tex]. When [tex]\( \sqrt{x-1} \)[/tex] = 0, [tex]\( f(x) = 0 \)[/tex]. As [tex]\( x \to \infty \)[/tex], [tex]\( \sqrt{x-1} \to \infty \)[/tex], making [tex]\( f(x) = 3 \sqrt{x-1} \to \infty \)[/tex]. Hence, the range of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex].
### 2. Important Points to Plot
Let's determine a few key points that lie on the function [tex]\( f(x) \)[/tex]:
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3 \sqrt{1 - 1} = 3 \times 0 = 0 \][/tex]
Point: [tex]\( (1, 0) \)[/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 3 \sqrt{2 - 1} = 3 \times 1 = 3 \][/tex]
Point: [tex]\( (2, 3) \)[/tex]
- When [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 3 \sqrt{4 - 1} = 3 \sqrt{3} \approx 3 \times 1.732 \approx 5.196 \][/tex]
Point: [tex]\( (4, 5.196) \)[/tex]
- When [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = 3 \sqrt{10 - 1} = 3 \sqrt{9} = 3 \times 3 = 9 \][/tex]
Point: [tex]\( (10, 9) \)[/tex]
### 3. Drawing the Graph
Using these points, we can sketch the graph of the function. Start by plotting the following points on the coordinate plane:
- [tex]\( (1, 0) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
- [tex]\( (4, 5.196) \)[/tex]
- [tex]\( (10, 9) \)[/tex]
These points are taken from the generated data:
[tex]\[ \begin{align*} (1.0, 0.0) & , (2.0, 3.0) \\ (4.0, 5.196) & , (10.0, 9.0) \end{align*} \][/tex]
### 4. Graph Shape Description
- The graph starts at [tex]\( (1, 0) \)[/tex] since [tex]\( f(1) = 0 \)[/tex].
- As [tex]\( x \)[/tex] increases, the function [tex]\( f(x) = 3 \sqrt{x-1} \)[/tex] grows rapidly but in a smooth, continuous curve.
- There is a gradual steepening as [tex]\( x \)[/tex] increases because the square root function increases more slowly than a linear function.
### Summary
The graph of [tex]\( f(x) = 3 \sqrt{x-1} \)[/tex] is a curve that begins at the point [tex]\( (1, 0) \)[/tex] and rises gradually. It represents a square root function scaled by a factor of 3, starting at [tex]\( x = 1 \)[/tex] and increasing continuously. The plotted points provide a guide to draw the function accurately.
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