If [tex]f(x)=\left\{\begin{array}{cc}x^2+2 & x \ \textless \ 0 \\ |x-2| & x \geq 0\end{array}\right.[/tex] then find
1) [tex]f ( 0 )=|0-2|=2[/tex]
2) [tex]f(-4)=(-4)^2+2=18[/tex]
3) [tex]f(1)=[/tex]

Evaluate f ( 0 ) using the definition f ( x ) = ∣ x − 2∣ since 0 l ess 0 , which gives f ( 0 ) = ∣0 − 2∣ = 2 .
Evaluate f ( − 4 ) using the definition f ( x ) = x 2 + 2 since − 4 < 0 , which gives f ( − 4 ) = ( − 4 ) 2 + 2 = 18 .
Evaluate f ( 1 ) using the definition f ( x ) = ∣ x − 2∣ since 1 l ess 0 , which gives f ( 1 ) = ∣1 − 2∣ = 1 .
The values of the function are f ( 0 ) = 2 , f ( − 4 ) = 18 , and f ( 1 ) = 1 , so the final answer is f ( 0 ) = 2 , f ( − 4 ) = 18 , f ( 1 ) = 1 .

Explanation

Understanding the Problem We are given a piecewise function f ( x ) and asked to evaluate it at three specific points: x = 0 , x = − 4 , and x = 1 . We need to use the correct part of the piecewise definition for each value of x .

Calculating f(0) For f ( 0 ) , since 0 l ess 0 (i.e., 0 l ess 0 is false), we use the second part of the definition, which is f ( x ) = ∣ x − 2∣ . Thus, f ( 0 ) = ∣0 − 2∣ = ∣ − 2∣ = 2 .

Calculating f(-4) For f ( − 4 ) , since − 4 < 0 , we use the first part of the definition, which is f ( x ) = x 2 + 2 . Thus, f ( − 4 ) = ( − 4 ) 2 + 2 = 16 + 2 = 18 .

Calculating f(1) For f ( 1 ) , since 1 l ess 0 (i.e., 1 l ess 0 is false), we use the second part of the definition, which is f ( x ) = ∣ x − 2∣ . Thus, f ( 1 ) = ∣1 − 2∣ = ∣ − 1∣ = 1 .

Final Answer Comparing our calculations with the provided solution, we see that f ( 0 ) = 2 and f ( − 4 ) = 18 are correct. However, the provided solution states f ( 1 ) = 3 , which is incorrect. The correct value is f ( 1 ) = 1 .


Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input. For example, income tax brackets are a piecewise function of income, where the tax rate changes at different income levels. Another example is the cost of shipping, which may have different rates depending on the weight of the package. Understanding how to evaluate piecewise functions is essential for working with these types of models.

Answered by GinnyAnswer | 2025-07-05

The evaluations for the piecewise function are: f ( 0 ) = 2 , f ( − 4 ) = 18 , and f ( 1 ) = 1 . Each value is determined by using the appropriate section of the function based on whether the input is less than or greater than zero. This method allows for accurate calculations based on piecewise definitions.
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Answered by Anonymous | 2025-07-07