Question

the graph represents the data cost for monthly internet service for a cell phone which function, ( c(x) ), represents the monthly cost in dollars in terms of ( x ), the number of gigabytes used in a month? ( c(x)=left{\begin{array}{ll}15, & 0 leq x leq 2 \\ 5 x+10, & 2<x leq 6 \\ 50, & 6<xend{array}
ight. ) ( c(x)=left{\begin{array}{ll}15, & 0 leq x<2 \\ 5 x+20, & 2 leq x<6 \\ 50, & 6 leq xend{array}
ight. ) ( c(x)=left{\begin{array}{ll}15 x, & 0 leq x leq 2 \\ 20 x+5, & 2<x leq 6 \\ 50 x, & 6<xend{array}
ight. ) ( c(x)=left{\begin{array}{ll}15 x, & 0 leq x<2 \\ 5 x+10, & 2 leq x<6 \\ 50 x, & 6 leq xend{array}
ight. )

Explanation:

Step1: Analyze the first interval ($0\leq x\leq2$)

When \(0\leq x\leq2\), from the graph, the cost \(C(x) = 15\) (a constant - horizontal line).

Step2: Analyze the second interval ($2 < x\leq6$)

We use the two - point form of a line. The two points on the line segment for \(2 < x\leq6\) are \((2,20)\) and \((6,40)\).
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{40 - 20}{6 - 2}=\frac{20}{4}=5\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(2,20)\), we get \(y-20 = 5(x - 2)\), which simplifies to \(y=5x+10\).

Step3: Analyze the third interval ($6 < x$)

When \(6 < x\), from the graph, the cost \(C(x)=50\) (a constant - horizontal line).

Answer:

\(C(x)=

$$\begin{cases}15, &0\leq x\leq2\\5x + 10, &2 < x\leq6\\50, &6 < x\end{cases}$$

\) (the first option)