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

Analyze the piecewise function for each given y value.
Solve for x in each piece of the function and check if the x value satisfies the condition for that piece.
If a valid x is found, the y value is in the range of the function.
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 a list of y values: -6, -4, -3, 0, 1, 3. We want to determine which of these y values are within the range of the function.

Checking y = -6 For y = − 6 :
If x < − 3 , then 2 x + 2 = − 6 . Solving for x , we get 2 x = − 8 , so x = − 4 . Since − 4 < − 3 , y = − 6 is in the range.

Checking y = -4 For y = − 4 :
If x < − 3 , then 2 x + 2 = − 4 . Solving for x , we get 2 x = − 6 , so x = − 3 . Since − 3 is not less than − 3 , this case doesn't work. If -3"> x > − 3 , then − x − 2 = − 4 . Solving for x , we get − x = − 2 , so x = 2 . Since -3"> 2 > − 3 , y = − 4 is in the range.

Checking y = -3 For y = − 3 :
If x = − 3 , then f ( x ) = x = − 3 . So y = − 3 is in the range.

Checking y = 0 For y = 0 :
If x < − 3 , then 2 x + 2 = 0 . Solving for x , we get 2 x = − 2 , so x = − 1 . Since − 1 is not less than − 3 , this case doesn't work. If -3"> x > − 3 , then − x − 2 = 0 . Solving for x , we get − x = 2 , so x = − 2 . Since − 2 is not greater than − 3 , this case doesn't work. However, let's consider -3"> x > − 3 . If x approaches − 3 from the right, then f ( x ) = − x − 2 approaches − ( − 3 ) − 2 = 3 − 2 = 1 . Since f ( x ) is continuous for -3"> x > − 3 , f ( x ) can take any value less than 1. Therefore, y = 0 is in the range.

Checking y = 1 For y = 1 :
If x < − 3 , then 2 x + 2 = 1 . Solving for x , we get 2 x = − 1 , so x = − 0.5 . Since − 0.5 is not less than − 3 , this case doesn't work. If -3"> x > − 3 , then − x − 2 = 1 . Solving for x , we get − x = 3 , so x = − 3 . Since − 3 is not greater than − 3 , this case doesn't work. So y = 1 is not in the range.

Checking y = 3 For y = 3 :
If x < − 3 , then 2 x + 2 = 3 . Solving for x , we get 2 x = 1 , so x = 0.5 . Since 0.5 is not less than − 3 , this case doesn't work. If -3"> x > − 3 , then − x − 2 = 3 . Solving for x , we get − x = 5 , so x = − 5 . Since − 5 is not greater than − 3 , this case doesn't work. So y = 3 is not in the range.

Final Answer Therefore, 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 the rules change based on the input. For example, consider a cell phone plan where you pay a fixed rate for the first 100 minutes and then a different rate for each additional minute. The cost can be modeled as a piecewise function. Similarly, income tax brackets are another example where the tax rate changes based on income levels. Understanding piecewise functions helps in analyzing and predicting outcomes in such scenarios.

Answered by GinnyAnswer | 2025-07-03

The values within the range of the piecewise-defined function are -6, -4, -3, and 0. Values 1 and 3 are not included in the range. Thus, the answer includes the four identified values.
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Answered by Anonymous | 2025-07-04