If $f(x)=\left\{\begin{array}{lll}x^2+1 & \text { if } & x \leq-1 \\ 2 x-3 & \text { if } & x>-1\end{array}\right.
$, what is $f(-2)$?
A. 4
B. 5
C. -7
D. -5
Answer (1)
Determine which case of the piecewise function to use based on the value of x .
Substitute x = − 2 into the correct expression, which is x 2 + 1 .
Calculate ( − 2 ) 2 + 1 .
The final answer is 5 .
Explanation
Understanding the Piecewise Function We are given a piecewise function f ( x ) and we want to find the value of f ( − 2 ) . The function is defined as:
-1\end{array}\right."> f ( x ) = { x 2 + 1 if x ≤ − 1 2 x − 3 if x > − 1
We need to determine which case to use based on the value of x .
Choosing the Correct Case Since we want to find f ( − 2 ) , we need to check if − 2 ≤ − 1 or -1"> − 2 > − 1 . Since − 2 is less than or equal to − 1 , we use the first case of the piecewise function:
f ( x ) = x 2 + 1
Substituting the Value Now, we substitute x = − 2 into the expression x 2 + 1 :
f ( − 2 ) = ( − 2 ) 2 + 1
Calculating the Result We calculate ( − 2 ) 2 which is ( − 2 ) × ( − 2 ) = 4 . Then we add 1:
f ( − 2 ) = 4 + 1 = 5
Final Answer Therefore, f ( − 2 ) = 5 .
Examples
Piecewise functions are used in real life to model situations where different rules apply 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 how to evaluate these functions is essential for making informed decisions in various scenarios.
