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The given problem involves the linear function:

\[
d(t) = 5 - 3t
\]

where \( d(t) \) represents the distance (in miles) from home and \( t \) represents the time (in hours) after leaving the library. Let's break down the problem step by step.

### Part (a): Sketching a Graph of the Function
The linear function \( d(t) = 5 - 3t \) is in slope-intercept form \( y = mx + b \), where:
- The slope \( m = -3 \) (this indicates the rate of change, meaning the distance decreases by 3 miles every hour).
- The y-intercept \( b = 5 \) (this represents the initial distance from home when \( t = 0 \), or the starting point).

To graph this function:
1. Plot the y-intercept at \( (0, 5) \).
2. Use the slope \( -3 \), which means for each increase of 1 in \( t \) (hours), the distance decreases by 3 miles.
3. You can calculate and plot another point, such as when \( t = 1 \), then \( d(1) = 5 - 3(1) = 2 \), so plot \( (1, 2) \).

### Part (b): Domain and Range
- **Domain:** The domain of \( d(t) \) represents the possible values of \( t \), which are the time values after leaving the library. Since the time \( t \) starts at 0 and is limited by when the distance reaches 0 (which happens when you arrive home), the domain is \( 0 \leq t \leq \frac{5}{3} \).
- **Range:** The range represents the possible values of \( d(t) \), the distance from home. The initial distance is 5 miles, and it decreases to 0 miles when you arrive home, so the range is \( 0 \leq d(t) \leq 5 \).

### Part (c): Intercepts of the Graph
- **Y-intercept (at \( t = 0 \)):** The function intersects the y-axis at \( (0, 5) \), meaning at the beginning (when \( t = 0 \)), you are 5 miles away from home.
- **X-intercept (when \( d(t) = 0 \)):** Set \( d(t) = 0 \) to find the x-intercept:
  \[
  0 = 5 - 3t \implies t = \frac{5}{3} \approx 1.67 \, \text{hours}.
  \]
  This means after about 1.67 hours, you will reach home (when the distance becomes 0).

Would you like me to provide a visual graph or further details?

---

### 5 Related Questions:
1. How would the graph change if the slope were steeper (e.g., \( d(t) = 5 - 4t \))?
2. Can you explain the meaning of a positive slope in this context?
3. How does the y-intercept change if you started farther from home?
4. What is the significance of the x-intercept in terms of distance and time?
5. How would the domain and range change if you biked home instead, traveling faster?

### Tip:
To graph a linear function, always start by identifying the y-intercept and slope, as these determine the overall shape and direction of the line.

The linear function d(t) = 5 - 3t gives your distance (d in miles) from home, t hours after you leave the library. a. Sketch a graph of the function. Write the domain and range in inequalities for d(t) in this context. b. What are the intercepts of the graph? Explain the meaning of any intercepts in this context.

Explore the linear function d(t) = 5 - 3t to understand how it describes distance from home over time. Learn to graph the function, find intercepts, and discuss the domain and range suitable for students in Grade 8-9. This example illustrates fundamental concepts in algebra including graphing linear equations and understanding intercepts.

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