Evaluate $f(3)$ for the piecewise function:
$f(x)=\left\{\begin{array}{l}
\frac{3 x}{2}+8, x<-6 \\
-3 x-2,-4 \leq x \leq 3 \\
4 x+4, x>3
\end{array}\right.$
Which value represents $f(3)$?
A. -11
B. 8
C. 12.5
D. 16
Answer (2)
Determine which piece of the piecewise function applies for x = 3 .
Since − 4 ≤ 3 ≤ 3 , the second piece, f ( x ) = − 3 x − 2 , applies.
Substitute x = 3 into the expression: f ( 3 ) = − 3 ( 3 ) − 2 .
Calculate the result: f ( 3 ) = − 11 , so the final answer is − 11 .
Explanation
Determine the relevant piece of the function We are given a piecewise function and asked to evaluate it at x = 3 . The function is defined as:
3 \end{array}\right."> f ( x ) = ⎩ ⎨ ⎧ 2 3 x + 8 , x < − 6 − 3 x − 2 , − 4 ≤ x ≤ 3 4 x + 4 , x > 3
To find f ( 3 ) , we need to determine which piece of the function applies when x = 3 .
Identify the correct interval Since − 4 ≤ 3 ≤ 3 , the second piece of the function applies, which is f ( x ) = − 3 x − 2 .
Substitute x=3 Now, we substitute x = 3 into the expression − 3 x − 2 :
f ( 3 ) = − 3 ( 3 ) − 2
Calculate f(3) Calculating the value:
f ( 3 ) = − 9 − 2 = − 11
State the final answer Therefore, f ( 3 ) = − 11 .
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 after that. Similarly, income tax brackets are a piecewise function, where the tax rate changes as your income increases. Understanding how to evaluate piecewise functions is essential for understanding these real-world scenarios.
To find f ( 3 ) for the piecewise function, we use the piece applicable for the interval − 4 ≤ 3 ≤ 3 , which is f ( x ) = − 3 x − 2 . Substituting x = 3 gives f ( 3 ) = − 11 . The chosen option is A. -11.
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