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. 9
C. 12.5
D. 18
Answer (2)
Determine that the second piece of the piecewise function applies for x = 3 .
Substitute x = 3 into the expression − 3 x − 2 .
Calculate f ( 3 ) = − 3 ( 3 ) − 2 = − 9 − 2 = − 11 .
The value of the function at x = 3 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 follows:
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 .
Calculate f(3) Now, we substitute x = 3 into the expression − 3 x − 2 :
f ( 3 ) = − 3 ( 3 ) − 2 = − 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 helps in calculating costs, taxes, and other real-world scenarios where different rules apply in different situations.
The value of the function at x = 3 is evaluated using the second piece of the piecewise function, which results in f ( 3 ) = − 11 . The correct choice is A. -11.
;
