Question 1.3.6: Evaluating a Piecewise Function Use the function that descri...
Evaluating a Piecewise Function
Use the function that describes the telephone plan
C(t)= \begin{cases}20 & \text { if } 0 \leq t \leq 60 \\ 20+0.40(t-60) & \text { if } t>60\end{cases}
to find and interpret each of the following:
a. C(30) b. C(100).
a. To find C(30), we let t = 30. Because 30 lies between 0 and 60 , we use the first line of the piecewise function.
C(t) = 20 This is the function’s equation for 0 ≤ t ≤ 60.
C(30) = 20 Replace t with 30. Regardless of this function’s input, the constant output is 20.
This means that with 30 calling minutes, the monthly cost is $20. This can be visually represented by the point (30, 20) on the first piece of the graph in Figure 1.37.
b. To find C(100), we let t = 100. Because 100 is greater than 60, we use the second line of the piecewise function.
\begin{array}{ll}C(t) = 20 + 0.40(t – 60)&\text{This is the function’s equation for t > 60.}\\[0.5 cm]C(100) = 20 + 0.40(100 – 60)&\text{Replace t with 100.}\\[0.5 cm]= 20 + 0.40(40)&\text{Subtract within parentheses: 100 – 60 = 40.}\\[0.5 cm]= 20 + 16&\text{Multiply: 0.40(40) = 16.}\\[0.5 cm]= 36&\text{Add: 20 + 16 = 36.}\end{array}
Thus, C(100)= 36. This means that with 100 calling minutes, the monthly cost is $36. This can be visually represented by the point (100, 36) on the second piece of the graph in Figure 1.37
