Question

Find a formula for a function describing the given situation. Graph the function and give a domain that makes sense for the problem. Recall that with constant speed. distance \(=\) speed \(\cdot\) time elapsed or \(d=v t\) A function \(y=f(x)\) such that if your car gets \(32 \mathrm{mi} / \mathrm{gal}\) and gasoline costs \(\$ x /\) gallon, then \(\$ 100\) is the cost of taking a \(y\) -mile trip

Short answer

Question: Determine the function that describes the cost of taking a 100-mile trip with a car that gets 32 miles per gallon when gasoline costs x dollars per gallon. Provide the graph's domain that makes sense for this problem. Answer: The function is \(y=f(x) = \frac{100x}{32}\), and the domain is \(0 \leq x \leq 10\).

Step-by-step solution

Write the cost equation for the car based on the given information

Since the car gets 32 miles per gallon and the gasoline costs x dollars per gallon, we can calculate the cost per mile by dividing the cost of gasoline by the miles per gallon: $$ \text{Cost per mile} = \frac{x \text{ dollars}}{32 \text{ miles}} $$

Define the function y = f(x)

To find the cost of taking a y-mile trip, we can multiply the cost per mile by the number of miles (y) in the trip: $$ f(x)=y=\text{Cost per mile} \times \text{Number of miles} = \frac{x}{32} \times 100 $$ This simplifies to: $$ f(x)=\frac{100x}{32} $$ So, the function \(y=f(x)\) represents the cost, in dollars, of taking a y-mile trip when gasoline costs x dollars per gallon.

Graph the function and provide a domain

To graph the function \(y=f(x) = \frac{100x}{32}\), we can notice that this is a linear function. Its slope is \(m=\frac{100}{32}\), which means that the cost of a trip, y, increases by \(100/32\) dollars for each dollar increase in the gasoline price per gallon, x. The domain for the problem should include only values of x that make sense for gasoline prices per gallon. Gas prices can't be negative, so x should be greater than or equal to 0. Since it's difficult to predict exactly how high gas prices could go, we'll use an arbitrary upper limit for the domain. For example, let's assume that gas prices won't go higher than \(10\) per gallon. This gives us a domain of \(0 \leq x \leq 10\). When graphing the function within this domain, we get a straight line that passes through the origin (0,0) and has a positive slope. In conclusion, the function \(y=f(x) = \frac{100x}{32}\) represents the cost of taking a y-mile trip when gasoline costs x dollars per gallon, with a domain of \(0 \leq x \leq 10\).

Key concepts

Linear Function

A linear function is a type of function that creates a straight line when graphed. These functions are defined by equations of the form \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. In simple terms, the slope \( m \) shows how much the value of \( y \) changes for each unit increase in \( x \).
In our given situation, the cost function is represented by \( y = f(x) = \frac{100x}{32} \). This means that for every dollar increase in the price of gasoline, the cost of the trip increases by \( \frac{100}{32} \).
Furthermore, there's no y-intercept here since when \( x = 0 \), the total cost is also zero, meaning the line passes through the origin (0,0). This feature is typical for problems where the cost or some measurement starts from zero when the input is zero.

Domain of a Function

The domain of a function refers to all possible input values (commonly denoted as \( x \)) that a function can accept. Determining the domain is crucial because it helps define the practical or real-world limits of the situation described by the function.
In the context of the cost function \( y = f(x) = \frac{100x}{32} \), the domain reflects the range of possible gasoline prices. Since gasoline can't have a negative price, the smallest value \( x \) can take is 0, translating to the lowest realistic price.
Additionally, although gasoline prices can fluctuate widely, it is reasonable to assume they will stay below a certain limit, such as 10 dollars per gallon. Hence, the domain for this function is \( 0 \leq x \leq 10 \). This ensures that our function only considers realistic and meaningful values.

Graphing a Function

Graphing a function involves plotting its equation on a coordinate plane, which helps visualize how the output values (\( y \)) change with different input values (\( x \)). For the linear function \( y = f(x) = \frac{100x}{32} \), the graph is a straight line.
When graphing linear functions, it is essential to identify the slope and intercepts. The slope here is positive, indicating the graph rises as \( x \) increases. Since the function lacks a numerical y-intercept, the line passes through the origin.
To properly graph this function within its domain, start at the origin (0,0) and use the slope to determine the next points. Increase \( x \) in small increments up to its maximum value of 10, calculating the corresponding \( y \) for each. This methodical plotting ensures the graph accurately reflects the function within the set domain, providing a clear picture of the relationship between the gasoline price and the trip cost.