Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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.
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
