Answer :
To rewrite the absolute value function [tex]\( f(x) = |-x + 4| \)[/tex] as a piecewise function, let's consider the definition and properties of the absolute value function. The absolute value function [tex]\(|a|\)[/tex] is defined as:
[tex]\[ |a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases} \][/tex]
Given the function [tex]\( f(x) = |-x + 4| \)[/tex], we need to analyze it under the conditions where the expression inside the absolute value, [tex]\(-x + 4\)[/tex], is non-negative and where it is negative.
1. Condition for [tex]\(-x + 4 \ge 0\)[/tex]:
Solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 4 \ge 0 \][/tex]
[tex]\[ -x \ge -4 \][/tex]
[tex]\[ x \le 4 \][/tex]
When [tex]\( x \le 4 \)[/tex], [tex]\(-x + 4\)[/tex] is non-negative, so the function can be written as:
[tex]\[ f(x) = -x + 4 \][/tex]
2. Condition for [tex]\(-x + 4 < 0\)[/tex]:
Solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 4 < 0 \][/tex]
[tex]\[ -x < -4 \][/tex]
[tex]\[ x > 4 \][/tex]
When [tex]\( x > 4 \)[/tex], [tex]\(-x + 4\)[/tex] is negative. In this case, we use the negative of the expression inside the absolute value, which is:
[tex]\[ f(x) = -(-x + 4) = x - 4 \][/tex]
Combining these two conditions, we can express the absolute value function [tex]\( f(x) = |-x + 4| \)[/tex] as a piecewise function:
[tex]\[ f(x) = \begin{cases} -x + 4 & \text{if } x \le 4 \\ x - 4 & \text{if } x > 4 \end{cases} \][/tex]
Next, let's plot the function [tex]\( f(x) \)[/tex] on the graph.
1. For [tex]\( x \leq 4 \)[/tex], the function [tex]\( f(x) = -x + 4 \)[/tex] is a linear function with a slope of [tex]\(-1\)[/tex] and a y-intercept at [tex]\( (0, 4) \)[/tex].
2. For [tex]\( x > 4 \)[/tex], the function [tex]\( f(x) = x - 4 \)[/tex] is a linear function with a slope of [tex]\( 1 \)[/tex] and a y-intercept at [tex]\( (4, 0) \)[/tex].
### Steps to draw the graph:
a. Draw the line segment for [tex]\( f(x) = -x + 4 \)[/tex] when [tex]\( x \leq 4 \)[/tex]:
- Start from the point [tex]\( (0, 4) \)[/tex].
- Move with a negative slope (-1) till [tex]\( x = 4 \)[/tex]:
- At [tex]\( x = 4 \)[/tex], [tex]\( f(4) = -4 + 4 = 0 \)[/tex].
- Point [tex]\( (4, 0) \)[/tex] marks the endpoint for this segment.
b. Draw the line segment for [tex]\( f(x) = x - 4 \)[/tex] when [tex]\( x > 4 \)[/tex]:
- Start from the point [tex]\( (4, 0) \)[/tex].
- Move with a positive slope (1):
- For example, at [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 5 - 4 = 1 \)[/tex].
- Continue the line onwards.
So, the graph will show a V-shape with the vertex at the point [tex]\( (4, 0) \)[/tex]. The left arm goes downwards from [tex]\( (0, 4) \)[/tex] to [tex]\( (4, 0) \)[/tex], and the right arm goes upwards from [tex]\( (4, 0) \)[/tex] extending to the right.
[tex]\[ |a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases} \][/tex]
Given the function [tex]\( f(x) = |-x + 4| \)[/tex], we need to analyze it under the conditions where the expression inside the absolute value, [tex]\(-x + 4\)[/tex], is non-negative and where it is negative.
1. Condition for [tex]\(-x + 4 \ge 0\)[/tex]:
Solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 4 \ge 0 \][/tex]
[tex]\[ -x \ge -4 \][/tex]
[tex]\[ x \le 4 \][/tex]
When [tex]\( x \le 4 \)[/tex], [tex]\(-x + 4\)[/tex] is non-negative, so the function can be written as:
[tex]\[ f(x) = -x + 4 \][/tex]
2. Condition for [tex]\(-x + 4 < 0\)[/tex]:
Solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 4 < 0 \][/tex]
[tex]\[ -x < -4 \][/tex]
[tex]\[ x > 4 \][/tex]
When [tex]\( x > 4 \)[/tex], [tex]\(-x + 4\)[/tex] is negative. In this case, we use the negative of the expression inside the absolute value, which is:
[tex]\[ f(x) = -(-x + 4) = x - 4 \][/tex]
Combining these two conditions, we can express the absolute value function [tex]\( f(x) = |-x + 4| \)[/tex] as a piecewise function:
[tex]\[ f(x) = \begin{cases} -x + 4 & \text{if } x \le 4 \\ x - 4 & \text{if } x > 4 \end{cases} \][/tex]
Next, let's plot the function [tex]\( f(x) \)[/tex] on the graph.
1. For [tex]\( x \leq 4 \)[/tex], the function [tex]\( f(x) = -x + 4 \)[/tex] is a linear function with a slope of [tex]\(-1\)[/tex] and a y-intercept at [tex]\( (0, 4) \)[/tex].
2. For [tex]\( x > 4 \)[/tex], the function [tex]\( f(x) = x - 4 \)[/tex] is a linear function with a slope of [tex]\( 1 \)[/tex] and a y-intercept at [tex]\( (4, 0) \)[/tex].
### Steps to draw the graph:
a. Draw the line segment for [tex]\( f(x) = -x + 4 \)[/tex] when [tex]\( x \leq 4 \)[/tex]:
- Start from the point [tex]\( (0, 4) \)[/tex].
- Move with a negative slope (-1) till [tex]\( x = 4 \)[/tex]:
- At [tex]\( x = 4 \)[/tex], [tex]\( f(4) = -4 + 4 = 0 \)[/tex].
- Point [tex]\( (4, 0) \)[/tex] marks the endpoint for this segment.
b. Draw the line segment for [tex]\( f(x) = x - 4 \)[/tex] when [tex]\( x > 4 \)[/tex]:
- Start from the point [tex]\( (4, 0) \)[/tex].
- Move with a positive slope (1):
- For example, at [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 5 - 4 = 1 \)[/tex].
- Continue the line onwards.
So, the graph will show a V-shape with the vertex at the point [tex]\( (4, 0) \)[/tex]. The left arm goes downwards from [tex]\( (0, 4) \)[/tex] to [tex]\( (4, 0) \)[/tex], and the right arm goes upwards from [tex]\( (4, 0) \)[/tex] extending to the right.
