1. **Problem Statement:** We are given two examples involving functions and asked to analyze their domain, range, and whether they are discrete or continuous. 2. **Example 1:** A student lives 4 miles from school and rides a bike at 1 mile per 6 minutes. The distance remaining $y$ is a function of time $x$ (in minutes). 3. **Function for Example 1:** The distance remaining decreases as time increases. The rate is 1 mile per 6 minutes, so the function is: $$y = 4 - \frac{1}{6}x$$ 4. **Domain and Range for Example 1:** - Domain: $0 \leq x \leq 24$ (since $4$ miles at $1$ mile per $6$ minutes means $4 \times 6 = 24$ minutes to reach school) - Range: $0 \leq y \leq 4$ 5. **Discrete or Continuous?** Since time and distance can take any value in the interval, this function is continuous. 6. **Example 2:** The function $y = 10 - 1.5x$ represents the total cost for John to ride $x$ rides at an amusement park. John has $10$ to spend. 7. **Domain and Range for Example 2:** - Domain: $x$ is discrete (number of rides), so $x \in \{0,1,2,\ldots,6\}$ because $10 - 1.5x \geq 0$ implies $x \leq \frac{10}{1.5} = 6.66$. - Range: Corresponding $y$ values from $10$ down to $0$ in steps of $1.5$. 8. **Discrete or Continuous?** Since $x$ represents number of rides (countable), this function is discrete. **Final answers:** - Example 1: Continuous function $y=4-\frac{1}{6}x$, domain $0 \leq x \leq 24$, range $0 \leq y \leq 4$. - Example 2: Discrete function $y=10-1.5x$, domain $x \in \{0,1,2,3,4,5,6\}$, range $y \in \{10,8.5,7,5.5,4,2.5,1\}$.