Suppose the piecewise function [tex]f(x)=\left\{\begin{array}{l}a \text { is shown below.: }\end{array}\right.[/tex] Enter the subfunction ([tex]f(x) \Rightarrow[/tex]) and subdomain for [tex]a[/tex] in order and separated by a comma. (Example: [tex]|x+3|-7, x\ \textless \ 2[/tex]) Enter the subdomain as a single equality or inequality.
Answer (2)
− x + 2 , 0 ≤ x ≤ 2
Explanation
Analyze the problem We are given a piecewise function f ( x ) and asked to find the subfunction and subdomain for the part labeled 'a'. By observing the graph, we can see that 'a' represents a line segment.
Find the slope The line segment 'a' passes through the points ( 0 , 2 ) and ( 2 , 0 ) . We can find the slope of the line using the formula m = x 2 − x 1 y 2 − y 1 . In this case, m = 2 − 0 0 − 2 = 2 − 2 = − 1 .
Find the equation Now we can use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , with the point ( 0 , 2 ) and slope m = − 1 . This gives us y − 2 = − 1 ( x − 0 ) , which simplifies to y = − x + 2 .
Determine the subdomain The line segment 'a' is defined for x values between 0 and 2, inclusive. Therefore, the subdomain is 0 ≤ x ≤ 2 .
Combine subfunction and subdomain Combining the subfunction and subdomain, we get − x + 2 , 0 ≤ x ≤ 2 .
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, a cell phone plan might charge one rate for the first 100 minutes of calls and a different rate for additional minutes. Similarly, a delivery service might charge a flat fee for packages up to a certain weight and an additional fee for heavier packages. Understanding piecewise functions helps in analyzing and predicting costs or outcomes in such scenarios.
The subfunction 'a' is identified as the linear equation − x + 2 with a subdomain of 0 ≤ x ≤ 2 . The line segment connects the points (0, 2) and (2, 0), yielding a slope of -1. Therefore, the complete answer representation is − x + 2 , 0 ≤ x ≤ 2 .
;
