Question: 1. A phone company offers a monthly data plan for $10 a month. The plan includes 2 megabytes of data, and charges $0.10 per megabytes above the 2 megabytes of data. Write a piecewise-defined function for $M(x)$, the cost for $x$ megabytes of data used in a month.
1. **State the problem:**
We need to write a piecewise-defined function $M(x)$ that represents the cost of a monthly data plan.
2. **Understand the pricing:**
- The base cost is $10$ for up to $2$ megabytes.
- For any data usage above $2$ megabytes, the cost increases by $0.10$ per megabyte.
3. **Write the piecewise function:**
- For $x \leq 2$, the cost is constant at $10$.
- For $x > 2$, the cost is $10$ plus $0.10$ times the amount of data above $2$ megabytes.
4. **Express mathematically:**
$$
M(x) = \begin{cases}
10 & \text{if } 0 \leq x \leq 2 \\
10 + 0.10(x - 2) & \text{if } x > 2
\end{cases}
$$
5. **Explain domain:**
Since data usage cannot be negative, the domain is $x \geq 0$.
6. **Final answer:**
$$
M(x) = \begin{cases}
10 & \text{if } 0 \leq x \leq 2 \\
10 + 0.10(x - 2) & \text{if } x > 2
\end{cases}
$$