f(x)=\left{\begin{array}{ll}3 x+4 & x \leq 2 \\ x^2+1 & x>2\end{array}\right. What is $f ( 3 )+ f ( 0 )$?
Answer (2)
To find f ( 3 ) + f ( 0 ) , we evaluate the piecewise function. f ( 3 ) = 10 from the second part and f ( 0 ) = 4 from the first part, resulting in a sum of 14 . Thus, the final answer is 14 .
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Evaluate f ( 3 ) using the second part of the piecewise function since 2"> 3 > 2 : f ( 3 ) = 3 2 + 1 = 10 .
Evaluate f ( 0 ) using the first part of the piecewise function since 0 ≤ 2 : f ( 0 ) = 3 ( 0 ) + 4 = 4 .
Add the two values: f ( 3 ) + f ( 0 ) = 10 + 4 = 14 .
The final answer is 14 .
Explanation
Understanding the Problem We are given a piecewise function and asked to find the sum of the function evaluated at two different points. The function is defined as: 2\end{array}\right."> f ( x ) = { 3 x + 4 x 2 + 1 x ≤ 2 x > 2 We need to find f ( 3 ) + f ( 0 ) .
Calculating f(3) First, let's find f ( 3 ) . Since 2"> 3 > 2 , we use the second part of the piecewise definition: f ( 3 ) = ( 3 ) 2 + 1 = 9 + 1 = 10. So, f ( 3 ) = 10 .
Calculating f(0) Next, let's find f ( 0 ) . Since 0 ≤ 2 , we use the first part of the piecewise definition: f ( 0 ) = 3 ( 0 ) + 4 = 0 + 4 = 4. So, f ( 0 ) = 4 .
Finding the Sum Now, we add the two values: f ( 3 ) + f ( 0 ) = 10 + 4 = 14. Therefore, f ( 3 ) + f ( 0 ) = 14 .
Examples
Piecewise functions are used in real life to model situations where different rules or formulas apply over different intervals. For example, tax brackets are a classic application of piecewise functions, where the tax rate changes based on income levels. Similarly, utility companies might use piecewise functions to calculate billing amounts, where the cost per unit of energy changes as consumption increases. Understanding how to evaluate and manipulate piecewise functions is essential for analyzing and predicting outcomes in these scenarios.
