Problem 89 - A Graphical Approach to Precalculus with Limits

Question

When a bolt of lightning strikes in the distance, there is often a delay between seeing the lightning and hearing the thunder. The function \(f(x)=\frac{x}{5}\) computes the approximate distance in miles between an observer and a bolt of lightning when the delay is \(x\) seconds. (a) Find \(f(15)\) and interpret the result. (b) Graph \(y=f(x) .\) Let the domain of \(f\) be \([0,20]\).

Step-by-Step Solution

Answer

f(15) = 3; the lightning is 3 miles away.

Step 1: Calculate f(15)

We need to calculate the value of the function \( f(x) = \frac{x}{5} \) when \( x = 15 \). Substitute \( 15 \) into the function:\[ f(15) = \frac{15}{5} \]Perform the division:\[ f(15) = 3 \]

Step 2: Interpret f(15)

The result \( f(15) = 3 \) indicates that when there is a 15-second delay between seeing lightning and hearing thunder, the lightning strike is approximately 3 miles away from the observer.

Step 3: Define the Domain for Graphing

The domain of the function \( f(x) \) is given as \([0, 20]\). This means we will plot the graph of the function from \( x = 0 \) to \( x = 20 \).

Step 4: Graph the Function y=f(x)

To graph \( y = f(x) = \frac{x}{5} \), plot points within the domain \([0, 20]\). For example, when \( x = 0 \), \( y = 0 \); when \( x = 5 \), \( y = 1 \); when \( x = 10 \), \( y = 2 \); when \( x = 15 \), \( y = 3 \); and when \( x = 20 \), \( y = 4 \). Connect these points with a straight line, which will be a linear graph showing a constant rate of change.

Key Concepts

Domain and Range

In the context of functions, the domain refers to all the possible input values, typically represented as "\(x\)," for which the function is defined. The range, on the other hand, consists of all possible outputs, represented as "\(f(x)\)". For the given function \(f(x) = \frac{x}{5}\), the domain has been specifically set as \([0, 20]\). This signifies that we are only interested in input values \(x\) from 0 to 20.

• Domain: \(x\) values from 0 to 20
• Range: Correspondingly, since \(f(x) = \frac{x}{5}\), the output changes linearly, giving us potential range values from 0 to 4.

Understanding the domain and range is crucial, especially in real-world applications. Here, the domain reflects the realistic range of delay times in seconds that an observer might note during a lightning event.

Linear Functions

Linear functions form a fundamental part of precalculus, representing relationships that have a constant rate of change. The function \(f(x) = \frac{x}{5}\) is a prime example of a linear function. In such functions, if you plot the input-output pairs on a graph, you'll see a straight line.

Key characteristics of linear functions include:

• Simplicity: Defined by equations of the form \(y = mx + b\), where \(m\) and \(b\) are constants. Here, \(m = \frac{1}{5}\) and \(b = 0\).
• Rate of Change: The coefficient \(m\) is significant, indicating the slope, or rate of change, in this scenario. A slope of \(\frac{1}{5}\) means for each unit increase in \(x\), \(f(x)\) increases by \(\frac{1}{5}\).

Recognizing and understanding linear functions can aid in predicting behavior, as they illustrate a consistent pattern, like the steady relationship between time delay and distance in a lightning strike scenario.

Graphing Functions

Graphing a function provides a visual representation of its behavior, showing how output values depend on input values. For \(f(x) = \frac{x}{5}\), graphing requires plotting points within the given domain \([0, 20]\).

Steps for graphing might include:

• Identify essential points from the function such as \((0, 0)\), \((5, 1)\), \((10, 2)\), etc., within the domain.
• Plot these points accurately on a coordinate system.
• Draw a straight line through these points, reflecting the linear nature of the function.

The graph of \(f(x) = \frac{x}{5}\) will be a straight line, with each point demonstrating the direct proportionality between time delay \(x\) and distance \(f(x)\). Understanding graphing is essential as it visually elucidates the function, assisting in the prediction and interpretation of a relationship in practical situations, such as this approximation of distance during lightning events.