Join the conversation on IDNLearn.com and get the answers you seek from experts. Our Q&A platform offers reliable and thorough answers to help you make informed decisions quickly and easily.
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 greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
