2. A hotel chain charges \( \$ 80 \) each night for the first two nights and \( \$ 45 \) for each additional night's stay T cost \( T \) is a function of the number of nights \( x \) that a guest stays. a. Complete the expressions in the following piecewise defined function.

If a guest stays for 1 or 2 nights, the cost \( T \) can be expressed as \( T(x) = 80x \) for \( 1 \leq x \leq 2 \). For stays longer than 2 nights, the cost includes the initial two nights at \( \$80 \) each, plus \( \$45 \) for each additional night. Therefore, for \( x > 2 \), the cost function becomes \( T(x) = 160 + 45(x - 2) \) or simplified, \( T(x) = 45x + 70 \). So, the completed piecewise function is: \[ T(x) = \begin{cases} 80x & \text{if } 1 \leq x \leq 2 \\ 160 + 45(x - 2) & \text{if } x > 2 \end{cases} \] The world of hotel pricing can be quite fun! The tiered pricing approach encourages guests to stay longer by providing an immediate discount for those who extend their stay beyond two nights. This strategy not only ensures a steady flow of guests but also enhances overall profitability. Just think about how many “extended vacation” deals you've come across! When dealing with piecewise functions, it's essential to pay attention to the defined intervals. A common mistake is to incorrectly apply the wrong formula for the number of nights. Always check which interval your stay falls into. Also, consider the implications of rounding your night counts; sometimes checking in late or checking out early affects costs, so keep those hotel rules in mind!