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To graph the piecewise function [tex]\( f(x) \)[/tex], we need to consider each part of the function separately and determine their values at specific points. Let's break it down step-by-step.
### 1. Understanding the Piecewise Function
The function is defined as follows:
- For [tex]\( x < 1 \)[/tex], the function is [tex]\( f(x) = 2x - 4 \)[/tex].
- For [tex]\( x \geq 1 \)[/tex], the function is [tex]\( f(x) = 1 + x \)[/tex].
### 2. Calculate Key Points
Let's find the values of the function at several points both below and above the boundary [tex]\( x = 1 \)[/tex].
#### For [tex]\( x < 1 \)[/tex]:
- When [tex]\( x = -2 \)[/tex]:
[tex]\[ f(x) = 2(-2) - 4 = -4 - 4 = -8 \][/tex]
So, the coordinate is [tex]\((-2, -8)\)[/tex].
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(x) = 2(0) - 4 = 0 - 4 = -4 \][/tex]
So, the coordinate is [tex]\((0, -4)\)[/tex].
#### Boundary at [tex]\( x = 1 \)[/tex]:
- When [tex]\( x = 1 \)[/tex] for the first piece [tex]\( 2x - 4 \)[/tex]:
[tex]\[ f(1) = 2(1) - 4 = 2 - 4 = -2 \][/tex]
- When [tex]\( x = 1 \)[/tex] for the second piece [tex]\( 1 + x \)[/tex]:
[tex]\[ f(1) = 1 + 1 = 2 \][/tex]
So, the coordinate is [tex]\((1, 2)\)[/tex].
#### For [tex]\( x \geq 1 \)[/tex]:
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ f(x) = 1 + 2 = 3 \][/tex]
So, the coordinate is [tex]\((2, 3)\)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ f(x) = 1 + 3 = 4 \][/tex]
So, the coordinate is [tex]\((3, 4)\)[/tex].
### 3. Plotting the Points
Now we can plot these points on the graph.
- For [tex]\( x < 1 \)[/tex]:
- [tex]\((-2, -8)\)[/tex]
- [tex]\((0, -4)\)[/tex]
- For [tex]\( x \geq 1 \)[/tex]:
- [tex]\((1, 2)\)[/tex]
- [tex]\((2, 3)\)[/tex]
- [tex]\((3, 4)\)[/tex]
### 4. Drawing the Graph
- Draw a line segment through the points [tex]\((-2, -8)\)[/tex] and [tex]\((0, -4)\)[/tex] for [tex]\( x < 1 \)[/tex]. This line should approach [tex]\((1, -2)\)[/tex] but not include it since [tex]\( x = 1 \)[/tex] is not in this part.
- At [tex]\( x = 1 \)[/tex], place an open circle at [tex]\((1, -2)\)[/tex] to indicate that this point is not included in the first piece.
- For [tex]\( x \geq 1 \)[/tex], start from the point [tex]\((1, 2)\)[/tex] and draw through the points [tex]\((2, 3)\)[/tex] and [tex]\((3, 4)\)[/tex]. This includes the point [tex]\((1, 2)\)[/tex] as a closed circle because it belongs to this part of the function.
### 5. Summary
- The line segment for [tex]\( 2x - 4 \)[/tex] starts from the left extending up to but not including [tex]\((1, -2)\)[/tex] (open circle).
- The line for [tex]\( 1 + x \)[/tex] starts from [tex]\((1, 2)\)[/tex] (closed circle) and extends to the right through [tex]\((2, 3)\)[/tex] and [tex]\((3, 4)\)[/tex].
This clearly illustrates the graph of the piecewise function [tex]\( f(x) \)[/tex].
### 1. Understanding the Piecewise Function
The function is defined as follows:
- For [tex]\( x < 1 \)[/tex], the function is [tex]\( f(x) = 2x - 4 \)[/tex].
- For [tex]\( x \geq 1 \)[/tex], the function is [tex]\( f(x) = 1 + x \)[/tex].
### 2. Calculate Key Points
Let's find the values of the function at several points both below and above the boundary [tex]\( x = 1 \)[/tex].
#### For [tex]\( x < 1 \)[/tex]:
- When [tex]\( x = -2 \)[/tex]:
[tex]\[ f(x) = 2(-2) - 4 = -4 - 4 = -8 \][/tex]
So, the coordinate is [tex]\((-2, -8)\)[/tex].
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(x) = 2(0) - 4 = 0 - 4 = -4 \][/tex]
So, the coordinate is [tex]\((0, -4)\)[/tex].
#### Boundary at [tex]\( x = 1 \)[/tex]:
- When [tex]\( x = 1 \)[/tex] for the first piece [tex]\( 2x - 4 \)[/tex]:
[tex]\[ f(1) = 2(1) - 4 = 2 - 4 = -2 \][/tex]
- When [tex]\( x = 1 \)[/tex] for the second piece [tex]\( 1 + x \)[/tex]:
[tex]\[ f(1) = 1 + 1 = 2 \][/tex]
So, the coordinate is [tex]\((1, 2)\)[/tex].
#### For [tex]\( x \geq 1 \)[/tex]:
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ f(x) = 1 + 2 = 3 \][/tex]
So, the coordinate is [tex]\((2, 3)\)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ f(x) = 1 + 3 = 4 \][/tex]
So, the coordinate is [tex]\((3, 4)\)[/tex].
### 3. Plotting the Points
Now we can plot these points on the graph.
- For [tex]\( x < 1 \)[/tex]:
- [tex]\((-2, -8)\)[/tex]
- [tex]\((0, -4)\)[/tex]
- For [tex]\( x \geq 1 \)[/tex]:
- [tex]\((1, 2)\)[/tex]
- [tex]\((2, 3)\)[/tex]
- [tex]\((3, 4)\)[/tex]
### 4. Drawing the Graph
- Draw a line segment through the points [tex]\((-2, -8)\)[/tex] and [tex]\((0, -4)\)[/tex] for [tex]\( x < 1 \)[/tex]. This line should approach [tex]\((1, -2)\)[/tex] but not include it since [tex]\( x = 1 \)[/tex] is not in this part.
- At [tex]\( x = 1 \)[/tex], place an open circle at [tex]\((1, -2)\)[/tex] to indicate that this point is not included in the first piece.
- For [tex]\( x \geq 1 \)[/tex], start from the point [tex]\((1, 2)\)[/tex] and draw through the points [tex]\((2, 3)\)[/tex] and [tex]\((3, 4)\)[/tex]. This includes the point [tex]\((1, 2)\)[/tex] as a closed circle because it belongs to this part of the function.
### 5. Summary
- The line segment for [tex]\( 2x - 4 \)[/tex] starts from the left extending up to but not including [tex]\((1, -2)\)[/tex] (open circle).
- The line for [tex]\( 1 + x \)[/tex] starts from [tex]\((1, 2)\)[/tex] (closed circle) and extends to the right through [tex]\((2, 3)\)[/tex] and [tex]\((3, 4)\)[/tex].
This clearly illustrates the graph of the piecewise function [tex]\( f(x) \)[/tex].
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