Answer :
To graph the given piecewise-defined function [tex]\( f(x) \)[/tex], follow these steps:
Piece 1: [tex]\( f(x) = -x + 3 \)[/tex] for [tex]\( x < 2 \)[/tex]
1. Expression: [tex]\( f(x) = -x + 3 \)[/tex]
- This is a linear function with a slope of -1 and a y-intercept at 3.
2. Graphing the Line:
- Start at the y-intercept (0, 3).
- Use the slope to find another point: from (0, 3), move down 1 unit and right 1 unit to get the point (1, 2).
- Continue plotting points along this line for values of [tex]\( x \)[/tex] less than 2.
3. Endpoint at [tex]\( x = 2 \)[/tex]:
- Since [tex]\( x = 2 \)[/tex] is not included in this interval (indicated by [tex]\( x < 2 \)[/tex]), draw an open circle at the point (2, 1). The value 1 is obtained by substituting [tex]\( x = 2 \)[/tex] into the expression [tex]\(-x + 3 \)[/tex], resulting in [tex]\( f(2) = -2 + 3 = 1 \)[/tex].
Piece 2: [tex]\( f(x) = 3 \)[/tex] for [tex]\( 2 \leq x < 4 \)[/tex]
1. Expression: [tex]\( f(x) = 3 \)[/tex]
- This is a constant function, meaning that [tex]\( f(x) \)[/tex] is always equal to 3 within the specified interval.
2. Graphing the Line:
- Draw a horizontal line at [tex]\( y = 3 \)[/tex] for the interval [tex]\( 2 \leq x < 4 \)[/tex].
3. Endpoints:
- At [tex]\( x = 2 \)[/tex], this section of the graph starts, and since the interval includes [tex]\( x = 2 \)[/tex] (indicated by [tex]\( \leq \)[/tex]), place a closed circle at (2, 3).
- At [tex]\( x = 4 \)[/tex], this section of the graph ends, and since the interval does not include [tex]\( x = 4 \)[/tex] (indicated by [tex]\( < 4 \)[/tex]), place an open circle at (4, 3).
Piece 3: [tex]\( f(x) = 4 - 2x \)[/tex] for [tex]\( x \geq 4 \)[/tex]
1. Expression: [tex]\( f(x) = 4 - 2x \)[/tex]
- This is a linear function with a slope of -2 and a y-intercept of 4.
2. Graphing the Line:
- Start at the point (4, -4) because substituting [tex]\( x = 4 \)[/tex] into the expression gives [tex]\( f(4) = 4 - 2*4 = 4 - 8 = -4 \)[/tex].
- Use the slope to find another point: from (4, -4), move down 2 units and right 1 unit to get the point (5, -6).
- Continue plotting points along this line for values of [tex]\( x \)[/tex] greater than or equal to 4.
3. Endpoint at [tex]\( x = 4 \)[/tex]:
- Since [tex]\( x = 4 \)[/tex] is included in this interval (indicated by [tex]\( x \geq 4 \)[/tex]), place a closed circle at the point (4, -4).
### Summary:
- For [tex]\( x < 2 \)[/tex]: Graph [tex]\( f(x) = -x + 3 \)[/tex]. Place an open circle at (2, 1) because x = 2 is not included.
- For [tex]\( 2 \leq x < 4 \)[/tex]: Graph [tex]\( f(x) = 3 \)[/tex]. Place a closed circle at (2, 3) and an open circle at (4, 3).
- For [tex]\( x \geq 4 \)[/tex]: Graph [tex]\( f(x) = 4 - 2x \)[/tex]. Place a closed circle at (4, -4).
By carefully plotting these pieces with their respective endpoints, you create an accurate graph of the piecewise function [tex]\( f(x) \)[/tex].
Piece 1: [tex]\( f(x) = -x + 3 \)[/tex] for [tex]\( x < 2 \)[/tex]
1. Expression: [tex]\( f(x) = -x + 3 \)[/tex]
- This is a linear function with a slope of -1 and a y-intercept at 3.
2. Graphing the Line:
- Start at the y-intercept (0, 3).
- Use the slope to find another point: from (0, 3), move down 1 unit and right 1 unit to get the point (1, 2).
- Continue plotting points along this line for values of [tex]\( x \)[/tex] less than 2.
3. Endpoint at [tex]\( x = 2 \)[/tex]:
- Since [tex]\( x = 2 \)[/tex] is not included in this interval (indicated by [tex]\( x < 2 \)[/tex]), draw an open circle at the point (2, 1). The value 1 is obtained by substituting [tex]\( x = 2 \)[/tex] into the expression [tex]\(-x + 3 \)[/tex], resulting in [tex]\( f(2) = -2 + 3 = 1 \)[/tex].
Piece 2: [tex]\( f(x) = 3 \)[/tex] for [tex]\( 2 \leq x < 4 \)[/tex]
1. Expression: [tex]\( f(x) = 3 \)[/tex]
- This is a constant function, meaning that [tex]\( f(x) \)[/tex] is always equal to 3 within the specified interval.
2. Graphing the Line:
- Draw a horizontal line at [tex]\( y = 3 \)[/tex] for the interval [tex]\( 2 \leq x < 4 \)[/tex].
3. Endpoints:
- At [tex]\( x = 2 \)[/tex], this section of the graph starts, and since the interval includes [tex]\( x = 2 \)[/tex] (indicated by [tex]\( \leq \)[/tex]), place a closed circle at (2, 3).
- At [tex]\( x = 4 \)[/tex], this section of the graph ends, and since the interval does not include [tex]\( x = 4 \)[/tex] (indicated by [tex]\( < 4 \)[/tex]), place an open circle at (4, 3).
Piece 3: [tex]\( f(x) = 4 - 2x \)[/tex] for [tex]\( x \geq 4 \)[/tex]
1. Expression: [tex]\( f(x) = 4 - 2x \)[/tex]
- This is a linear function with a slope of -2 and a y-intercept of 4.
2. Graphing the Line:
- Start at the point (4, -4) because substituting [tex]\( x = 4 \)[/tex] into the expression gives [tex]\( f(4) = 4 - 2*4 = 4 - 8 = -4 \)[/tex].
- Use the slope to find another point: from (4, -4), move down 2 units and right 1 unit to get the point (5, -6).
- Continue plotting points along this line for values of [tex]\( x \)[/tex] greater than or equal to 4.
3. Endpoint at [tex]\( x = 4 \)[/tex]:
- Since [tex]\( x = 4 \)[/tex] is included in this interval (indicated by [tex]\( x \geq 4 \)[/tex]), place a closed circle at the point (4, -4).
### Summary:
- For [tex]\( x < 2 \)[/tex]: Graph [tex]\( f(x) = -x + 3 \)[/tex]. Place an open circle at (2, 1) because x = 2 is not included.
- For [tex]\( 2 \leq x < 4 \)[/tex]: Graph [tex]\( f(x) = 3 \)[/tex]. Place a closed circle at (2, 3) and an open circle at (4, 3).
- For [tex]\( x \geq 4 \)[/tex]: Graph [tex]\( f(x) = 4 - 2x \)[/tex]. Place a closed circle at (4, -4).
By carefully plotting these pieces with their respective endpoints, you create an accurate graph of the piecewise function [tex]\( f(x) \)[/tex].
