In this section, we will introduce a topic that will be essential for continued mathematical learning: functions. Functions should be thought of as machines that turn one number into another number, much like a cash register can turn a number of pounds of fruit into a price.
We are familiar with the symbol. This symbol is used to turn numbers into their square roots. Sometimes itβs simple to do this on paper or in our heads, and sometimes it helps to have a calculator. We can see some calculations in Figure 2.
The symbol represents a process; itβs a way for us to turn numbers into other numbers. This idea of having a process for turning numbers into other numbers is the fundamental topic of this chapter.
A function is a process for turning numbers into (potentially) different numbers. The process must be consistent, in that whenever you apply it to some particular number, you always get the same result.
Section 5 covers a more technical definition for functions, and covers topics that are more appropriate when using that definition. Definition 3 is so broad that you probably use functions all the time.
The function is consistent; for example, every time you evaluate , you always get 3. One interesting fact is that is not found on most keyboards, and yet computers can still find square roots. Computer technicians write when they want to compute a square root, as we see in Figure 5.
The parentheses in are very important. To see why, try to put yourself in the βmindβ of a computer. The computer will recognize sqrt and know that it needs to compute a square root but without parentheses it will think that it needs to compute sqrt4 and then put a 9 on the end, which would produce a final result of . This is probably not what was intended. And so the purpose of the parentheses in sqrt(49) is to be deliberately clear.
Functions have their own names. Weβve seen a function named , but any name you can imagine is allowable. In the sciences, it is common to name functions with whole words, like or . In math, we often abbreviate such function names to or . And of course, since the word βfunctionβ itself starts with βf,β we will often name a function .
In some contexts, the symbol might represent a variable (a number that is represented by a letter) and in other contexts, might represent a function (a process for changing numbers into other numbers). By staying conscious of the context of an investigation, we avoid confusion.
is pronounced βf of 13.β The word βofβ is very important, because it reminds us that is a process and we are about to apply that process to the input value . So is the function, is the input, and is the output weβd get from using as input.
is pronounced βf of x.β This is just like the previous example, except that the input is not any specific number. The value of could be or any other number. Whatever βs value, means the corresponding output from the function .
is pronounced βBudgetDeficit of 2017.β This is probably about a function that takes a year as input, and gives that yearβs federal budget deficit as output. The process here of changing a year into a dollar amount might not involve any mathematical formula, but rather looking up information from the Congressional Budget Officeβs website.
While a function has a name like , and the input to that function often has a variable name like , the expression represents the output of the function. To be clear, is not a function. Rather, is a function, and its output when the number was used as input.
In the following examples, a function is given using a formula, and we will evaluate the function at specific values. See Section 1.1 for a review on evaluating expressions.
As mentioned in Warning 6, we need to remain conscious of the context of any symbol we are using. Consider the expression . This could easily mean the output of a function with input . It could also mean that two numbers and need to be multiplied. It all depends on the context in which these symbols are being used.
Sometimes itβs helpful to think of a function as a machine, as in Figure 16. A function has the capacity to take in all kinds of different numbers into itβs hopper (feeding tray) as inputs and transform them into their outputs.
Since functions are potentially complicated, we want ways to understand them more easily. Two basic tools for understanding a function better are tables and graphs.
Consider the function , that takes a year as its input and outputs the US federal budget deficit for that year. For example, the Congressional Budget Officeβs website tells us that is trillion. If weβd like to understand this function better, we might make a table of all the inputs and outputs we can find. Using the CBOβs website (www.cbo.gov/topics/budget), we can put together Table 18.
These observations help us understand a little better. For instance, based on these observations which do you think is larger: the difference between and , or the difference between and ?
Another powerful tool for understanding a function better is a graph. Given a function , one way to make its graph is to take a table of input and output values, and read each row as the coordinates of a point in the -plane.
Returning to the function that we studied in Example 17, in order to make a graph of this function we view Table 18 as a list of points with and coordinates, as in Figure 24. We then plot these points on a set of coordinate axes, as in Figure 25. The points have been connected with a curve so that we can see the overall pattern given by the progression of points. Since there was not any actual data for inputs in between any two years, the curve is dashed. That is, this curve is dashed because it just represents someoneβs best guess as to how to connect the plotted points. Only the plotted points themselves are precise.
How has this graph helped us to understand the function better? All of the observations that we made in Example 17 are perhaps even more clear now. For instance, the spike in the deficit between 2008 and 2009 is now visually apparent. Seeking an explanation for this spike, we recall that there was a financial crisis in late 2008. Revenue from income taxes dropped at the same time that federal money was spent to prevent further losses.
Just as in the previous example, weβve plotted points where we have concrete coordinates, and then we have made our best attempt to connect those points with a curve. Unlike the previous example, here we believe that points will continue to follow the same pattern indefinitely to the right, and so we have added an arrowhead to the graph.
What has this graph done to improve our understanding of ? As inputs (-values) increase, the outputs (-values) increase too, although not at the same rate. In fact we can see that our graph is steep on its left, and less steep as we move to the right. This confirms our earlier observation in Example 20 that outputs increase by smaller and smaller amounts as the input increases.
Given a function , when we refer to a graph of we are not referring to an entire picture, like Figure 28. A graph of is only part of that pictureβthe curve and the points that it connects. Everything else (axes, tick marks, the grid, labels, and the surrounding white space) is just useful decoration so that we can read the graph more easily.
It is common to refer to the graph of as the graph of the equation . However, we should avoid saying βthe graph of .β That would indicate a misunderstanding of our notation. Since is the output for a certain input . That means that is just a number and not worthy of a two-dimensional picture.
While it is important to be able to make a graph of a function , we also need to be capable of looking at a graph and reading it well. A graph of provides us with helpful specific information about ; it tells us what does to its input values. When we were making graphs, we plotted points of the form
In Figure 32 we have a graph of a function . If we wish to find , we recognize that is being used as an input. So we would want to find a point of the form . Seeking out -coordinate in Figure 32, we find that the only such point is . Therefore the output for is ; in other words .
Suppose that is the unemployment function of time. That is, is the unemployment rate in the United States in year . The graph of the equation is given in Figure 35 (data.bls.gov/timeseries/LNS14000000).
What was the unemployment in 2008? It is a straightforward matter to use Figure 35 to find that unemployment was almost in 2008. Asking this question is exactly the same thing as asking to find . That is, we have one question that can either be asked in an everyday-English way or which can be asked in a terse, mathematical notation-heavy way:
If we use the table to establish that , then we should be prepared to translate that into everyday-English using the context of the function: In 2009, unemployment in the U.S. was about .
If we ask the question βwhen was unemployment at ,β we can read the graph and see that there were two such times: mid-2007 and about 2016. But there is again a more mathematical notation-heavy way to ask this question. Namely, since we are being told that the output of is , we are being asked to solve the equation . So the following communicate the same thing:
We have noted that functions are complicated, and we want ways to make them easier to understand. Itβs common to find a problem involving a function and not know how to find a solution to that problem. Most functions have at least four standard ways to think about them, and if we learn how to translate between these four perspectives, we often find that one of them makes a given problem easier to solve.
Consider a function that squares its input and then adds . Translate this verbal description of into a table, a graph, and a formula.
Explanation.
To make a table for , weβll have to select some input -values. These choices are left entirely up to us, so we might as well choose small, easy-to-work-with values. However we shouldnβt shy away from negative input values. Given the verbal description, we should be able to compute a column of output values. Figure 39 is one possible table that we might end up with.
Figure11.1.39.
Once we have a table for , we can make a graph for as in Figure 40, using the table to plot points.
Figure11.1.40.
Lastly, we must find a formula for . This means we need to write an algebraic expression that says the same thing about as the verbal description, the table, and the graph. For this example, we can focus on the verbal description. Since takes its input, squares it, and adds , we have that
Let be the function that takes a Celsius temperature as input and outputs the corresponding Fahrenheit temperature. Translate this verbal description of into a table, a graph, and a formula.
Explanation.
To make a table for , we will need to rely on what we know about Celsius and Fahrenheit temperatures. It is a fact that the freezing temperature of water at sea level is 0 Β°C, which equals 32 Β°F. Also, the boiling temperature of water at sea level is 100 Β°C, which is the same as 212 Β°F. One more piece of information we might have is that standard human body temperature is 37 Β°C, or 98.6 Β°F. All of this is compiled in Figure 42. Note that we tabulated inputs and outputs by working with the context of the function, not with any computations.
Figure11.1.42.
Once a table is established, making a graph by plotting points is a simple matter, as in Figure 43. The three plotted points seem to be in a straight line, so we think it is reasonable to connect them in that way.
Figure11.1.43.
To find a formula for , the verbal definition is not of much direct help. But βs graph does seem to be a straight line. And linear equations are familiar to us. This line has a -intercept at and a slope we can calculate: . So the equation of this line is . On the other hand, the equation of this graph is , since it is a graph of the function . So evidently,
You have a savings account at a bank which accrues some interest over time. The formula models the amount of money, in dollars, in the savings account months after it was initially setup.
A seafood company exports from Portland, Oregon to local restaurants in the city. They want to begin exporting outside of Portland. The formula models the cost, in dollars, to export seafood miles away from Portland.
A group of scientists are studying the relationship between the salinity (salt concentration) and the oxygen concentration in the Columbia River Estuary. The formula models the oxygen concentration, as a percentage, when the salinity in the estuary is at percent.
A group of engineers are studying the effects of temperature on the lifetime of a specific type of battery. The formula models the lifetime, in years, of the battery if it is kept in an environment with a temperature of degrees Celsius.
Sharell started saving in a piggy bank on her birthday. The function models the amount of money, in dollars, in Sharellβs piggy bank. The independent variable represents the number of days passed since her birthday.
Julie will spend to purchase some bowls and some plates. Each bowl costs , and each plate costs . The function models the number of plates Julie will purchase, where represents the number of bowls Julie will purchase.
Irene will spend to purchase some bowls and some plates. Each plate costs , and each bowl costs . The function models the number of bowls Irene will purchase, where represents the number of plates to be purchased.
Suppose that is the function that computes how many miles are in feet. Find the formula for . If you do not know how many feet are in one mile, you can look it up on Google.
Suppose that is the function that computes how many kilograms are in pounds. Find the formula for . If you do not know how many pounds are in one kilogram, you can look it up on Google.
Suppose that is the function that the phone company uses to determine what your bill will be (in dollars) for a long-distance phone call that lasts minutes. Each call costs a fixed price of $ plus cents per minute. Write a formula for this linear function .
Suppose that is the function that gives the total cost (in dollars) of downhill skiing times during a season with a $500 season pass. Write a formula for . [@KeyboardInstructions(βYou need to write the entire formula, including [|f(x)=|]*.)@]**
Using function notation, express the carβs position after hours. The answer here is not a formula, itβs just something using function notation like f(8).
Where is the car then? The answer here is a number with units.
Use function notation to express the question, βWhen is the car going ?β The answer is an equation that uses function notation; something like f(x)=23. You are not being asked to actually solve the equation, just to write down the equation.
Where is the car when it is going ? The answer here is a number with units. You are being asked a question about its position, but have been given information about its speed.
Using function notation, express the carβs position after hours. The answer here is not a formula, itβs just something using function notation like f(8).
Where is the car then? The answer here is a number with units.
Use function notation to express the question, βWhen is the car going ?β The answer is an equation that uses function notation; something like f(x)=23. You are not being asked to actually solve the equation, just to write down the equation.
Where is the car when it is going ? The answer here is a number with units. You are being asked a question about its position, but have been given information about its speed.
Describe your own example of a function that has real context to it. You will need some kind of input variable, like βnumber of years since 2000β or βweight of the passengers in my car.β You will need a process for using that number to bring about a different kind of number. The process does not need to involve a formula; a verbal description would be great, as would a formula.
The following figure has the graph , which models a particleβs distance from the starting line in feet, where stands for time in seconds since timing started.
The following figure has the graph , which models a particleβs distance from the starting line in feet, where stands for time in seconds since timing started.
Use the given graph of a function , along with , and to answer the following questions. Some answers are points, and should be entered as ordered pairs. Some answers ask you to solve for , so the answer should be in the form x=...
