Which values are within the range of the piecewise-defined function?
[tex]f(x)=\left\{\begin{array}{ll}2 x+2, & x\ \textless \ -3 \\ x, & x=-3 \\ -x-2, & x\ \textgreater \ -3\end{array}\right.[/tex]
y=-6
y=-4
y=-3
y=0
y=1
y=3
Answer (2)
Check if y = − 6 : It is in the range.
Check if y = − 4 : It is in the range.
Check if y = − 3 : It is in the range.
Check if y = 0 : It is in the range.
Check if y = 1 : It is not in the range.
Check if y = 3 : It is not in the range.
Therefore, the values within the range are − 6 , − 4 , − 3 , 0 .
Explanation
Understanding the Problem We are given a piecewise-defined function: -3 \end{array}\right."> f ( x ) = ⎩ ⎨ ⎧ 2 x + 2 , x , − x − 2 , x < − 3 x = − 3 x > − 3 and we need to determine which of the following y values are within the range of the function: -6, -4, -3, 0, 1, 3.
Checking Each y Value We need to check each y value to see if there exists an x value such that f ( x ) = y . We will consider each piece of the function separately.
Checking All Cases
y = − 6 :
If 2 x + 2 = − 6 and x < − 3 , then 2 x = − 8 , so x = − 4 . Since − 4 < − 3 , y = − 6 is in the range.
If x = − 3 , then y = − 3 , which is not -6.
If − x − 2 = − 6 and -3"> x > − 3 , then − x = − 4 , so x = 4 . Since -3"> 4 > − 3 , y = − 6 is in the range.
y = − 4 :
If 2 x + 2 = − 4 and x < − 3 , then 2 x = − 6 , so x = − 3 . But we need x < − 3 , so this case doesn't work.
If x = − 3 , then y = − 3 , which is not -4.
If − x − 2 = − 4 and -3"> x > − 3 , then − x = − 2 , so x = 2 . Since -3"> 2 > − 3 , y = − 4 is in the range.
y = − 3 :
If 2 x + 2 = − 3 and x < − 3 , then 2 x = − 5 , so x = − 2.5 . But we need x < − 3 , so this case doesn't work.
If x = − 3 , then y = − 3 . So y = − 3 is in the range.
If − x − 2 = − 3 and -3"> x > − 3 , then − x = − 1 , so x = 1 . Since -3"> 1 > − 3 , y = − 3 is in the range.
y = 0 :
If 2 x + 2 = 0 and x < − 3 , then 2 x = − 2 , so x = − 1 . But we need x < − 3 , so this case doesn't work.
If x = − 3 , then y = − 3 , which is not 0.
If − x − 2 = 0 and -3"> x > − 3 , then − x = 2 , so x = − 2 . Since -3"> − 2 > − 3 , y = 0 is in the range.
y = 1 :
If 2 x + 2 = 1 and x < − 3 , then 2 x = − 1 , so x = − 0.5 . But we need x < − 3 , so this case doesn't work.
If x = − 3 , then y = − 3 , which is not 1.
If − x − 2 = 1 and -3"> x > − 3 , then − x = 3 , so x = − 3 . But we need -3"> x > − 3 , so this case doesn't work.
y = 3 :
If 2 x + 2 = 3 and x < − 3 , then 2 x = 1 , so x = 0.5 . But we need x < − 3 , so this case doesn't work.
If x = − 3 , then y = − 3 , which is not 3.
If − x − 2 = 3 and -3"> x > − 3 , then − x = 5 , so x = − 5 . But we need -3"> x > − 3 , so this case doesn't work.
Final Answer Based on the above analysis, the values within the range of the piecewise-defined function are -6, -4, -3, and 0.
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 a fixed rate for the first 1000 minutes and a different rate for each additional minute. Similarly, income tax brackets are defined using a piecewise function, where the tax rate changes as income increases. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
The values within the range of the piecewise-defined function are -6, -4, -3, and 0. The values 1 and 3 are not included in the range. Therefore, the chosen values are -6, -4, -3, and 0.
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