A contagious disease whose spread is unchecked can devastate a confined population. For example, in the early 16th-century, Spanish troops introduced smallpox into the Aztec population in Central America, and the resulting epidemic contributed significantly to the fall of Montezuma’s empire.
Suppose that an outbreak of cholera follows severe flooding in an isolated town of 5000 people. Initially (Day 0), 40 people are infected. Every day after that, 25% of those still healthy fall ill.
At the beginning of the first day (Day 1), how many people are still healthy? How many will fall ill during the first day? What is the total number of people infected after the first day?
Check your results against the table below. Subtract the total number of infected residents from 5000 to find the number of healthy residents at the beginning of the second day. Then fill in the rest of the table for 10 days. (Round off decimal results to the nearest whole number.)
Use the last column of the table to plot the total number of infected residents, \(I\text{,}\) against time, \(t\text{,}\) on the grid. Connect your data points with a smooth curve.
Do the values of \(I\) approach some largest value? Draw a dotted horizontal line at that value of \(I\text{.}\) Will the values of \(I\) ever exceed that value?
Look back at the table. What is happening to the number of new patients each day as time goes on? How is this phenomenon reflected in the graph? How would your graph look if the number of new patients every day were a constant?
Summarize your work: In your own words, describe how the number of residents infected with cholera changes with time. Include a description of your graph.
We often want to predict values of one variable from the values of a related variable. For example, when a physician prescribes a drug in a certain dosage, she needs to know how long the dose will remain in the bloodstream. A sales manager needs to know how the price of his product will affect its sales. A function is a special type of relationship between variables that allows us to make such predictions.
We have already seen some examples of functions. For instance, suppose it costs $800 for flying lessons, plus $30 per hour to rent a plane. If we let \(C\) represent the total cost for \(t\) hours of flying lessons, then
The variable \(t\) is called the input variable, and \(C\) is the output variable. Given a value of the input, we can calculate the corresponding output value using the formula for the function. Thus, for example
We can display the relationship between two variables by a table or by ordered pairs. The input variable is the first component of the ordered pair, and the output variable is the second component. For the example above we have:
A function is a relationship between two variables for which exactly one value of the output variable is determined by each value of the input variable.
The distance, \(d\text{,}\) traveled by a car in 2 hours is a function of its speed, \(r\text{.}\) If we know the speed of the car, we can determine the distance it travels by the formula \(d = r \cdot 2\text{.}\)
The cost of a fill-up with unleaded gasoline is a function of the number of gallons purchased. The gas pump represents the function by displaying the corresponding values of the input variable (number of gallons) and the output variable (cost).
Score on the Scholastic Aptitude Test (SAT) is not a function of score on an IQ test, because two people with the same score on an IQ test may score differently on the SAT; that is, a person’s score on the SAT is not uniquely determined by his or her score on an IQ test.
As part of a project to improve the success rate of freshmen, the counseling department studied the grades earned by a group of students in English and algebra. Do you think that a student’s grade in algebra is a function of his or her grade in English? Explain why or why not.
Phatburger features a soda bar, where you can serve your own soft drinks in any size. Do you think that the number of calories in a serving of Zap Kola is a function of the number of fluid ounces? Explain why or why not.
A function can be described in several different ways. In the following examples, we consider functions defined by tables, by graphs, and by equations.
When we use a table to describe a function, the first variable in the table (the left column of a vertical table or the top row of a horizontal table) is the input variable, and the second variable is the output. We say that the output variable is a function of the input.
The table below shows data on sales compiled over several years by the accounting office for Eau Claire Auto Parts, a division of Major Motors. In this example, the year is the input variable, and total sales is the output. We say that total sales, \(S\text{,}\)is a function of\(t\text{.}\)
If we know the weight of the article being mailed, we can find the postage from the table. For instance, a catalog weighing 4.5 ounces would require $1.56 in postage. In this example, \(w\) is the input variable and \(p\) is the output variable. We say that \(p\)is a function of\(w\text{.}\)
According to these data, cholesterol count is not a function of age, because several patients who are the same age have different cholesterol levels. For example, three different patients are 51 years old but have cholesterol counts of 227, 209, and 216, respectively. Thus, we cannot determine a unique value of the output variable (cholesterol count) from the value of the input variable (age). Other factors besides age must influence a person’s cholesterol count.
Note that several different inputs for a function can have the same output. For example, the inputs 4.5 and 4.25 in part (b) of Example 5.1.5 above have output $1.56. However, a single input cannot have more than one output, as illustrated in part (c) of the Example.
The graph shows the number of hours, \(H\text{,}\) that the sun is above the horizon in Peoria, Illinois, on day \(t\text{,}\) where \(t = 0\) on January 1.
The input variable, \(t\text{,}\) appears on the horizontal axis. The number of daylight hours, \(H\text{,}\) is a function of the date. The output variable appears on the vertical axis.
The point on the curve where \(t = 150\) has \(H \approx 14.1\text{,}\) so Peoria gets about 14.1 hours of daylight when \(t = 150\text{,}\) which is at the end of May.
The maximum value of 14.4 hours occurs on the longest day of the year, when \(t \approx 170\text{,}\) about three weeks into June. The minimum of 9.6 hours occurs on the shortest day, when \(t \approx 355\text{,}\) about three weeks into December.
The graph shows the elevation in feet, \(a\text{,}\) of the Los Angeles Marathon course at a distance \(d\) miles into the race. (Source: Los Angeles Times, March 3, 2005)
As of 2020, One World Trade Center in New York City is the nation’s tallest building, at 1776 feet. If an algebra book is dropped from the top of One World Trade Center, its height above the ground after \(t\) seconds is given by the equation
\begin{equation*}
h = 1776 - 16t^2
\end{equation*}
Thus, after \(\alert{1}\) second the book’s height is
For this function, \(t\) is the input variable and \(h\) is the output variable. For any value of \(t\text{,}\) a unique value of \(h\) can be determined from the equation for \(h\text{.}\) We say that \(h\)is a function of\(t\text{.}\)
There is a convenient notation for discussing functions. First, we choose a letter, such as \(f\text{,}\)\(g\text{,}\) or \(h\) (or \(F\text{,}\)\(G\text{,}\) or \(H\)), to name a particular function. (We can use any letter, but these are the most common choices.)
For instance, in Example 5.1.11, the height, \(h\text{,}\) of a falling algebra book is a function of the elapsed time, \(t\text{.}\) We might call this function \(f\text{.}\) In other words, \(f\) is the name of the relationship between the variables \(h\) and \(t\text{.}\) We write
\begin{equation*}
h = f (t)
\end{equation*}
which means "\(h\) is a function of \(t\text{,}\) and \(f\) is the name of the function."
The new symbol \(f(t)\text{,}\) read "\(f\) of \(t\text{,}\)" is another name for the height, \(h\text{.}\) The parentheses in the symbol \(f(t)\) do not indicate multiplication. (It would not make sense to multiply the name of a function by a variable.) Think of the symbol \(f(t)\) as a single variable that represents the output value of the function.
Perhaps it seems complicated to introduce a new symbol for \(h\text{,}\) but the notation \(f(t)\) is very useful for showing the correspondence between specific values of the variables \(h\) and \(t\text{.}\)
\begin{equation*}
f(1)=1760 \quad\text{ and } \quad f(2)=1712
\end{equation*}
which we read as "\(f\) of 1 equals 1760" and "\(f\) of 2 equals 1712." The values for the input variable, \(t\text{,}\) appear inside the parentheses, and the values for the output variable, \(h\text{,}\) appear on the other side of the equation.
We can also find the input (or inputs) corresponding to a given output. For example, if \(p=g(w)\) is the postage function, we solve the equation \(g(w)=0.84\) by finding all input values, \(w\text{,}\) that correspond to the output $0.84. According to the table in Example 5.1.5 part (b), any value of \(w\) greater than 1 but less than or equal to 2 is a solution.
When you exercise, your heart rate should increase until it reaches your target heart rate. The table shows target heart rate, \(r = f(a) \text{,}\) as a function of age.
To evaluate a function described by an equation, we simply substitute the given input value into the equation to find the corresponding output, or function value.
Complete the table displaying ordered pairs for the function \(f(x) = 5 - x^3\text{.}\) Evaluate the function to find the corresponding \(f(x)\)-value for each value of \(x\text{.}\)
The graph shows \(C=h(t)\text{,}\) where \(C\) stands for the number of customers (in thousands) signed up for a new movie streaming service, measured in months after their advertising campaign at \(t=0\) in January.
The graph shows \(P\) as a function of \(t\text{.}\)\(P\) is the number of houses in Cedar Grove who have had solar panels installed \(t\) years after 2000.
The graph shows the U.S. unemployment rate, \(U=F(t)\text{,}\) where \(t\text{,}\) represents years. Give your answers to the questions below in function notation. (Source: U.S. Bureau of Labor Statistics)
The graph shows the federal minimum wage, \(M\text{,}\) over the past five decades, adjusted for inflation to reflect its buying power in 2004 dollars. (Source: www.infoplease.com)
What is the largest function value on the graph, and when did it occur? Write your answer with function notation, and explain what it means about the federal minimum wage.
Give two years in which the minimum wage was worth $8 in 2004 dollars. Does this fact mean that \(M\) is not a function of \(t\text{?}\) Why or why not?
The bar graph shows the percent of Earth’s surface that lies at various altitudes or depths below the surface of the oceans. (Depths are given as negative altitudes.) (Source: Open University)
Energy is necessary to raise the temperature of a substance, and it is also needed to melt a solid substance to a liquid. The table shows data from heating a solid sample of stearic acid. Heat was applied at a constant rate throughout the experiment.
