According to Wikipedia, a function has the following definition:

“In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.”

If you actually read the entire Wikipedia entry you will find the definition of a function can be quite lengthy in order to cover all the universe of mathematics. In this activity we are going to keep the language simple, informal and we will use some analogies and examples to arrive at a working definition of functions.

Start by remebmering this: one input, one output.

A function is a rule that takes a piece of “usable” information that you pick (input) and gives you some kind of result (output). By “usable” we mean information that has meaning for the function.

Think of a function as a sort of machine. Using some input, the machine does something to the input and then out pops a result. In this course our inputs will be mostly numbers and the process will yeild numbers as results.

This brings up some good questions:

• How do we communicate that we have a function? There has to be a simple or quick way to communicate we have a function.

• What kinds of inputs can the function use? Using 24 hour time, you can choose to go outside at \(10\) hours or \(15\) hours. But you can't go outside at \(37\) hours because there are only \(24\) hours in a day. The input of \(37\) hours is not usable.

• What are the possible predicted temperatures throughout the day? Is any temperature possible or is there a window of reasonable temperature expectations?

• How does the temperature outside change as time changes throughout the day? It may get hotter or colder, does the temperature change quickly or slowly. We need a way to communicate how the temperature behaves during a day.

We start with notation. We don't want to keep writing the word, F U N C T I O N, all the time. That's just way too many letters.

Math is just another language. We use it to communicate measurements, changes in value and how things are related to or affect one another. Almost everything you say or describe in Math, you can say or describe in English.

In this next exercise we learn to communicate the position of a train as time passes and develop the notation to express where the train is at different times. We will assume that this train runs on time, every day and the conditions every day are the same. You know, an alternate universe called “example land”.

In English you can communicate information about time and position in a simple sentence like

\(2\) hours after 8 a.m. the train is \(30\) miles from downtown.

Here is what that same sentence looks like in Spanish.

\(2\) horas despues de las 8 de la mañana el tren esta 30 millas lejos del centro.

Here is what that same sentence could look like in “math”.

\(f(2) = 30\)

The math “sentence” above is actually an equation and it uses function notation to express the information about the train's position at a particular time.

In math we don't really have subjects or verbs or objects. Instead we have input values (numbers) and we have output values. The function is the “machine” or the rule that tells us how to use the input to make an output.

Variables are used to represent values we don't know or that may change. The distance from downtown is constantly changing with time, so we can let \(D\) represent the “Train's distance in miles from downtown”.

For the “number of hours after 8 a.m.”, we can use the variable \(t\) where t counts how many hours have passed since 8 a.m.

Typically we use variables that have something to do with the context of the problem, but sometimes we generalize an expression or equation with \(x\)'s and \(y\)'s when we have nothing specific in mind.

Since the position of the train depends on the amount of time that has passed, we say “the distance depends on time”. In fact, the distance of the train from downtown is a function of time.

Function notation allows us to communicate that distance is a function of time by simply writing \(D = f(t)\text{.}\) The notation \(f(t)\) is read as, “\(f\) of \(t\)”.

Remember \(t\) is a value. The function uses \(t\) to create the value \(D\text{.}\)

Also, don't confuse the notation with multiplication! \(f(t)\) does not mean \(f\) times \(t\text{.}\)

Function notation like \(f(t)\) or \(f(x)\) or \(f(\text{anything})\) implies there is some kind of relation, some kind of rule, that takes a piece of information and returns a response or result. The format of function notation works on the basic principle of input and output and we often use the notation when we refer to things we don't yet know.

For instance, if you work as a health assistant, you probably earn some hourly salary and you probably work some number of hours each week. I have no idea how many hours a week you work and no idea how much you earn. But I do know your weekly salary is related to the number of hours you work. In fact, your weekly salary is a function of the number of hours you work.

If \(t\) is the number of hours you work and \(S\) is your weekly salary, I can use function notation to say, \(S = f(t)\) where \(f(t)\) is the function, the instructions, that calculates your weekly salary. If you work \(12\) hours, then \(t = 12\) so \(f(12)\) is your weekly salary for working \(12\) hours.

Notice I still don't know what the salary actually is, but I can use function notation to talk about it. \(f(12)\) literally means, “whatever you earn when you work \(12\) hours.”

In our train example we chose some input, a time, and the function gives an output of distance. The notation tells the reader what the input is by showing it in the parentheses.\begin{equation*}f(\text{input}) = \mathit{output}\end{equation*} The function notation \(f(0)\) is telling us the input is \(0\text{.}\) That means, “\(0\) hours after 8 a.m.” In other words it's 8 a.m.

On the other hand, \(f(0)\) represents an output. It is “the train's distance from downtown \(0\) hours after 8 a.m.”. But we don't know what that distance actually is yet. \(f(0)\) in this case means, “What ever the distance is at t = 0”.

Nonetheless, the input and output create an ordered pair \((0, f(0))\text{.}\) On a graph the ordered pair is a point.

Here is another example to further explore the use function notation.

When two or more variables are related to each other, the formula or rule that relates them is called a relation. There is a relation between age and height, you can make ordered pairs of \((\text{age}, \text{height})\) for different people and plot the points on a graph. You might even find a formula that predicts height based on age (or vice versa?)

In every relation you have encountered so far in this chapter, each input gives you only one output. Earlier, for the train example we had the function \(D = f(t)\text{.}\) At any given time, like \(t = 10\text{,}\) the train can only be in one place, \(f(10)\text{.}\) It's not possible for the train to be in two or more places at the same time.

Also notice that each input has its own unique output. Every day, at \(t = 10\) the train is always in the same place, \(f(10) = 30\) (it's a very dependable train). These two conditions make the relation a function. Postion is a function of time.

But not all relations are functions. Just because two things are related does not mean one is a function of the other.

Twice each day the train is \(30\) miles away from downtown. Once in the morning at on its way out, then later in the afternoon on the way back into town. If we choose postion as input and time as output we realize that the input \(D = 30\) gives us two answers: one time in the morning and another in the afternoon. Thus time is not a function of position. All we have is a relation.

Tables are one way to express inputs and outputs of a relation. In a vertical table the inputs are on the left while the outputs are on the right. In a horizontal table, the top row is the input and the bottom row is the output.

When looking at a graph it's easy to tell if the graph is that of a function or not. Remember the inputs are found along the horizontal axis while the outputs are found along the vertical axis. The rule is the same: one input, one unique output.

The previous example leads us to a simple test we can use to determine if a graph is a function or not. It's called the vertical line test.

Given a graph, draw a vertical line through any place on the graph. If the vertical line crosses the graph at most one time (no matter where you draw the line), then the graph is a funciton. If the vertical line crosses the graph anywhere at more than one point, then the graph is not a function.

It's one thing to talk about “The train's distance from downtown \(2\) hours after 8 a.m.” and a completely different thing to say what that distance actually is.

Evaluate means to find the value of the output given an input value. Therefore, evaluate \(f(2)\text{,}\) in our train example means “Find the value of the train's distance from downtown \(2\) hours after 8 a.m.”

To evaluate \(f(2)\) we can use a graph, a table or a formula if they are avaible.

The important thing is we need some time input value to determine the output value for the train's distance from downtown. Also, for each input the function yields only one result and each individual input always gives the same single result.

In general when the inputs and outputs are nothing in particular, we use \(x\) as input and \(y\) as output so that \(y = f(x)\text{.}\) We use “\(x\)” as symbol to hold a spot for the actual input value while “\(y\)” is a spot holder for whatever the output value will be. The \(x\) basically says, “When you get the input, put it here” in all the places where the \(x\) is found.

Consider the function \(f(x) = 50x + 10\text{.}\) The function has no context (no story) so we won't bother using units for either \(x\) or \(y\text{.}\)

If \(x = 1\) then we can evaluate \(f(1)\) by replacing the “\(x\)” in the formula with a \(1\text{.}\) Literally, \(f(1)\) means, “Whatever you get when you put \(1\) into the formula”.

Similarly, if you change the input to \(x = 2\) then you can evaluate the function \(f(2)\text{,}\) where \(f(2)\) represents, “Whatever you get when you put \(2\) into the formula”.

Function notation is a way to tell the reader information about the input and how to use it to generate an output. Again, the way it works is \(f(\text{input})=\text{output}\text{.}\) Whatever is in the parenthesis is the input.

The input can be a simple number like \(2\) or \(-4\) or the input can be a complicated expression like \(x^2 - x\text{.}\)

Your success in this course is almost entirely dependent on your understanding of this simple concept: The “blank” method.

When you see a formula like \(f(x) = x^2 - x\text{.}\) It means this:\begin{equation*}f(\text{blank}) = (\text{blank})^2 - (\text{blank})\end{equation*}

At least that's the way formulas really work.

Using this function as an example, \(f(5)\) means “The formula with a \(5\) in it”. Literally it means place number \(5\) in all the blanks.

The beauty of this is it does not matter what you put into the formula (as long as the formula can handle it which we will get into later), the formula does the same thing every time.

You can put \(5\) into the formula or you can put something complicated into the formula like \(\frac{x^2}{x^2 + 1}\) which would look like this\begin{equation*}f\mathopen{}\left(\frac{x^2}{x^2 + 1}\right)\mathclose{} = \mathopen{}\left(\frac{x^2}{x^2 + 1}\right)^2\mathclose{} - \mathopen{}\left(\frac{x^2}{x^2 + 1}\right)\mathclose{}\end{equation*}

Simplifying this equation is another matter.

What if the output, the result, is already known and we want to know all the inputs that give us this result? This is called solve.

If you live in Oregon in the winter, you might ask, “What time will the temperature outside be \(37\) degrees fahrenheit?” Typically there might be two answers to this question, one time in the morning and another time in the afternoon.

You could answer the question with actual times if you had a time vs. temperature graph or a formula or a table of times and temperatures.

Solve means to work backwards from a known or given output value and determine all the inputs, if any, that give us the desired output.

Suppose we want the output of a function to be \(y =0\text{.}\) Therefore, we must find all the input values of \(x\) that gives us that result. There may not be any, but there could be one, several or even infinite possible inputs that gives \(y = 0\) as the output.

In English we could ask the question for the example in the previous exercise,

What input makes the output equal to \(0\text{?}\)

There would be three answers:\begin{equation*}x = - 4 , -1\text{ or }2\end{equation*}

All of these values of \(x\) give us the same result of \(y = 0\text{.}\)

In math we could ask the same question, but it comes out as a statement.

Solve \(f(x) = 0\)

In other words, “Find the input that gives you a particular result”. In our case the result we want is an output value of \(0\text{.}\) There are still three answers to the question and each answer is called a “solution”.

The statement:

Solve \(f(x) = 0\)

has three solutions: \begin{equation*}x = -4, -1 \text{ or }2\end{equation*}

If we look at functions graphically, the math sentence

Solve the equation \(f(x) = K\)

where \(K\) is some number, means to “find all the \(x\)-values where the graph of the function reaches a height of \(K\) units high.” this equation may have one, several, or infinite or even no solutions.

Even equations where a formula equals another formula can be solved graphically. Consider \begin{equation*}2^x + 0.025 = 0.55x - 0.3\end{equation*}To solve graphically, make a graph on your graphing calculator, then find the \(x\) values where the graphs have the same output (height). In other words, find where the graphs intersect.

Often equations are too difficult or even impossible to solve using algebra. But if you can graph the equations on a graphing calculator or on the web, you can solve graphically to get at least approximate solutions.