Cost of a Hotel Stay A hotel chain charges 75 each night...

cost of a Hotel Stay A hotel chain charges $\$ 75$ each night for the first two nights and $\$ 50$ for each additional night's stay. The total 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. $$ T(x)=\left\{\begin{array}{ll} & \text { if } 0 \leq x \leq 2 \\ & \text { if } x>2 \end{array}\right. $$ b. Find $T(2), T(3),$ and $T(5)$ c. What do your answers in part (b) represent?

cost of a Hotel Stay A hotel chain charges $\$ 75$ each night for the first two nights and $\$ 50$ for each additional night's stay. The total 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.
$$
T(x)=\left\{\begin{array}{ll}
& \text { if } 0 \leq x \leq 2 \\
& \text { if } x>2
\end{array}\right.
$$
b. Find $T(2), T(3),$ and $T(5)$
c. What do your answers in part (b) represent?

Key Concepts

Piecewise Defined Functions

A piecewise defined function is one that uses different formulas or rules for different parts of its domain. In many real-world problems, such as cost calculations where the pricing rules change based on conditions, the function is naturally broken into separate cases, each valid over a specified interval. This concept helps in organizing and computing values for inputs that fall into distinct categories.

Function Evaluation

Function evaluation involves substituting specific input values into a function to determine the corresponding output. In the context of the problem, evaluating the function at various points demonstrates how the overall rule applies to specific scenarios. This process is essential for verifying that the model correctly captures the behavior intended in different segments of the domain.

Mathematical Modeling of Real-World Situations

Mathematical modeling is the process of representing a real-world scenario through mathematical expressions or functions. In this case, the cost of a hotel stay is modeled by a function that changes its structure depending on the number of nights stayed, reflecting the business rule of different pricing for the initial nights and additional nights. This concept illustrates how mathematical tools can be applied to simulate and analyze practical situations using precise rules and calculations.

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