Consider the function:
[tex]f(x)=\left\{\begin{array}{cc}\frac{7}{2}+2 x, & x \leq-1 \\-5+\frac{3 x}{2}, & -1 < x < 3 \\\frac{1}{4} x, & x \geq 3\end{array}\right.[/tex]
What are these values?
[tex]f(-3)=[/tex]
[tex]f(-1)=[/tex]
[tex]f(3)=[/tex]
Answer (2)
Evaluate f ( − 3 ) using the first piece of the function since − 3 ≤ − 1 : f ( − 3 ) = 2 7 + 2 ( − 3 ) = − 2 5 .
Evaluate f ( − 1 ) using the first piece of the function since − 1 ≤ − 1 : f ( − 1 ) = 2 7 + 2 ( − 1 ) = 2 3 .
Evaluate f ( 3 ) using the third piece of the function since 3 ≥ 3 : f ( 3 ) = 4 1 ( 3 ) = 4 3 .
The values are f ( − 3 ) = − 2 5 , f ( − 1 ) = 2 3 , and f ( 3 ) = 4 3 , so the final answer is f ( − 3 ) = − 2.5 , f ( − 1 ) = 1.5 , f ( 3 ) = 0.75 .
Explanation
Understanding the Problem We are given a piecewise function f ( x ) and asked to evaluate it at three specific points: x = − 3 , x = − 1 , and x = 3 . We need to determine which piece of the function applies to each value of x and then substitute the value into the corresponding expression.
Calculating f(-3) For f ( − 3 ) , since − 3 ≤ − 1 , we use the first piece of the function: f ( x ) = 2 7 + 2 x . Substituting x = − 3 , we get f ( − 3 ) = 2 7 + 2 ( − 3 ) = 2 7 − 6 = 2 7 − 2 12 = − 2 5 = − 2.5.
Calculating f(-1) For f ( − 1 ) , since − 1 ≤ − 1 , we again use the first piece of the function: f ( x ) = 2 7 + 2 x . Substituting x = − 1 , we get f ( − 1 ) = 2 7 + 2 ( − 1 ) = 2 7 − 2 = 2 7 − 2 4 = 2 3 = 1.5.
Calculating f(3) For f ( 3 ) , since 3 ≥ 3 , we use the third piece of the function: f ( x ) = 4 1 x . Substituting x = 3 , we get f ( 3 ) = 4 1 ( 3 ) = 4 3 = 0.75.
Final Answer Therefore, we have found the values of the function at the specified points: f ( − 3 ) = − 2.5 , f ( − 1 ) = 1.5 , and f ( 3 ) = 0.75 .
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 one rate for the first 100 minutes of calls and a different rate for each additional minute. Similarly, income tax brackets are a form of piecewise function, where the tax rate changes depending on the income level. Understanding how to evaluate piecewise functions is essential for analyzing and predicting outcomes in these scenarios.
The evaluations of the piecewise function are as follows: f ( − 3 ) = − 2.5 , f ( − 1 ) = 1.5 , and f ( 3 ) = 0.75 . Each value was derived from the appropriate piece of the function based on the given conditions for x .
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