A piecewise function is defined as shown.
[tex]f(x)=\left\{\begin{array}{cc}-x, & x \leq-1 \\ 1, & x=0 \\ x+1 & x \geq 1\end{array}\right.[/tex]
What is the value of [tex]f(x)[/tex] when [tex]x=3[/tex]?
A. -3
B. -2
C. 1
D. 4
Answer (2)
Determine which condition x = 3 satisfies in the piecewise function.
Since 3 ≥ 1 , use the third case of the piecewise function: f ( x ) = x + 1 .
Substitute x = 3 into f ( x ) = x + 1 to find f ( 3 ) .
Calculate f ( 3 ) = 3 + 1 = 4 , so the final answer is 4 .
Explanation
Understanding the Problem We are given a piecewise function and asked to find the value of f ( x ) when x = 3 . The piecewise function is defined as: f ( x ) = ⎩ ⎨ ⎧ − x , 1 , x + 1 x ≤ − 1 x = 0 x ≥ 1
Choosing the Correct Piece To find f ( 3 ) , we need to determine which condition x = 3 satisfies. We have three conditions: x ≤ − 1 , x = 0 , and x ≥ 1 . Since 3 ≥ 1 , we use the third case of the piecewise function, which is f ( x ) = x + 1 .
Calculating f(3) Now, we substitute x = 3 into f ( x ) = x + 1 to find f ( 3 ) : f ( 3 ) = 3 + 1 = 4
Final Answer Therefore, the value of f ( x ) when x = 3 is 4.
Examples
Piecewise functions are used in real life to model situations where the rules change based on the input. For example, a cell phone plan might charge one rate for the first 100 minutes and a different rate for each minute thereafter. Similarly, income tax brackets are a form of piecewise function, where the tax rate changes as income increases. Understanding piecewise functions helps in analyzing and predicting outcomes in such scenarios.
When evaluating the piecewise function at x = 3, we find that it fits the condition x \geq 1. Using this condition, we substitute 3 into the function and calculate f(3) to be 4.
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