Question

write a piecewise function that models the cellular phone billing plan described below. let x represent the number of minutes used and c(x) represent the cost for those x minutes. then graph the function. $20.00 per month buys 350 minutes. additional time costs $0.25 per minute. c(x) = \

$$\begin{cases} 20 & \\text{if } 0 \\leq x \\leq 350 \\\\ 0.25x - 67.5 & \\text{if } x > 350 \\end{cases}$$

(simplify your answer. do not include the $ symbol in your answer.) choose the correct graph of the function. \bigcirc a. graph \bigcirc b. graph \bigcirc c. graph \bigcirc d. graph

Explanation:

Step1: Analyze the piecewise function

The function \( C(x) \) has two parts:

• For \( 0 \leq x \leq 350 \), \( C(x) = 20 \). This is a horizontal line (constant function) because the cost is fixed at 20 for minutes from 0 to 350.
• For \( x > 350 \), \( C(x)=0.25x - 67.5 \). Let's check the value at \( x = 350 \) for this part: \( 0.25(350)-67.5=87.5 - 67.5 = 20 \), which matches the first part at \( x = 350 \), so the function is continuous at \( x = 350 \). The slope of this line is \( 0.25 \), which is positive, so it's an increasing line for \( x>350 \).

Step2: Analyze the graphs

• Graph A: It has a non - constant part before \( x = 350 \), which does not match \( C(x)=20 \) for \( 0\leq x\leq350 \).
• Graph B: It has a non - horizontal part for \( 0\leq x\leq350 \), which is incorrect as \( C(x) \) should be 20 (horizontal) in that interval.
• Graph C: For \( 0\leq x\leq350 \), it is a horizontal line (constant at 20), and for \( x > 350 \), it is an increasing line (since the slope of \( 0.25x-67.5 \) is positive). This matches our analysis of the piecewise function.
• Graph D: The part for \( x>350 \) does not seem to have the correct slope or starting point (the connection at \( x = 350 \) is not as per the function's value).

Answer:

C