Select the correct answer from each drop-down menu.Consider the function [tex]$f(x)=\left|\frac{-x-3}{2}\right|-1$[/tex].The piecewise functions that coincide with the given function are[tex]$\frac{(x+3)}{2}-1$[/tex] when [tex]$x$[/tex] is greater than or equal to -3, [tex]$\frac{-(x+3)}{2}-1$[/tex] when [tex]$x$[/tex] is less than -3.
Answer (2)
The function f ( x ) = ∣ 2 − x − 3 ∣ − 1 is split into cases based on the sign of 2 − x − 3 .
When x ≤ − 3 , f ( x ) = − 2 x + 5 .
When -3"> x > − 3 , f ( x ) = 2 x + 1 .
The piecewise functions that coincide with the given function are 2 x + 1 when -3"> x > − 3 and − 2 x + 5 when x ≤ − 3 .
Explanation
Analyze the absolute value expression We are given the function f ( x ) = 2 − x − 3 − 1 and we want to express it as a piecewise function. The key is to analyze the expression inside the absolute value, 2 − x − 3 , and determine when it is positive or negative.
Case 1: x <= -3 The expression 2 − x − 3 is non-negative when − x − 3 ≥ 0 , which simplifies to − x ≥ 3 , or x ≤ − 3 . When x ≤ − 3 , we have 2 − x − 3 = 2 − x − 3 . Therefore, f ( x ) = 2 − x − 3 − 1 = 2 − x − 3 − 2 = 2 − x − 5 = − 2 x + 5 .
Case 2: x > -3 The expression 2 − x − 3 is negative when − x − 3 < 0 , which simplifies to − x < 3 , or -3"> x > − 3 . When -3"> x > − 3 , we have 2 − x − 3 = − 2 − x − 3 = 2 x + 3 . Therefore, f ( x ) = 2 x + 3 − 1 = 2 x + 3 − 2 = 2 x + 1 .
Piecewise Function Thus, the piecewise function is given by -3 \end{cases}"> f ( x ) = { − 2 x + 5 , x ≤ − 3 2 x + 1 , x > − 3
Matching the Piecewise Function Now, let's compare this to the given options. We have:
2 x + 3 − 1 = 2 x + 1 when x ≥ 3 . This doesn't match our piecewise function.
2 x − 3 − 1 = 2 x − 5 when x < 3 . This doesn't match our piecewise function.
2 x + 3 − 1 = 2 x + 1 when -3"> x > − 3 . This matches our piecewise function for -3"> x > − 3 .
− 2 x + 3 − 1 = − 2 x + 5 when x ≥ − 3 . This doesn't quite match our piecewise function, since our function is − 2 x + 5 when x ≤ − 3 .
− 2 x + 3 − 1 = − 2 x + 5 when x is less than or equal to − 3 . This matches our piecewise function for x ≤ − 3 .
Examples
Piecewise functions are used in real life to model situations where the rules change based on the input. For example, consider a cell phone plan where you pay a fixed rate for a certain amount of data, and then a different rate for any data used beyond that amount. The cost of the plan can be modeled as a piecewise function, where one piece represents the cost within the initial data allowance, and the other piece represents the cost for exceeding that allowance. Understanding piecewise functions helps in analyzing and predicting costs in such scenarios.
The function f ( x ) = 2 − x − 3 − 1 is represented as a piecewise function: -3 \end{cases}"> f ( x ) = { − 2 x + 5 , 2 x + 1 , x ≤ − 3 x > − 3 . The appropriate matches from the given functions are 2 x + 1 for -3"> x > − 3 and − 2 x + 5 for x ≤ − 3 .
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