ObjectivesStudent Learning Outcomes
After completing this activity you will be able to:
Interpret the meaning of function notation.
Use function notation to describe an event.
Evaluate a function and solve an equation using function notation.
After completing this activity you will be able to:
Interpret the meaning of function notation.
Use function notation to describe an event.
Evaluate a function and solve an equation using function notation.
In this activity we define a function and develop an understanding of function notation by comparing statements we already understand in English to equivalent statements written in “math” language.
The prerequisites for this lesson are knowing how to read a graph or a table of values and how to use a formula to make a calculation.
The mathematical concept of a function can be extremely complicated. It took thousands of years to come up with it, define it and formalize it. For example, just look at this definition from Wikipedia (do not spend too much time on it.)
You could spend your whole life learning about what a function is and never get to the rest of this book! For this reason we will keep things simple and informal in order to attain a quick and accessible understanding of topics, including functions. Students continuing in mathematics will add to their knowledge of functions as needed and as part of a life-long process.
Start by remembering this: one input, one output.
A function is a rule that takes a piece of “usable” information that you choose (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 yield numbers as results.
A simple example of a function is a temperature converter. It converts a temperature from degrees Fahrenheit to degrees Celsius.
The function is the rule or process used to convert from one unit of temperature to the other.
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? The temperature function in the previous exercise could only accept temperatures between \(-60\) and \(110\) degrees Fahrenheit. The input of \(150\) degrees \(F\) is not usable because it is not available to choose.
What are the possible output temperatures in degrees Celsius? Is any temperature possible or is there a window of reasonable temperature expectations?
How are the two temperature units related? If you increase degrees F, does the temperature in degrees \(C\) also increase?
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. For a function, the input is called the independent variable, and the output is called the dependent variable. This is because the output depends on the input.
The distance from downtown is constantly changing with time, so we can let the dependent variable \(D\) represent the “Train's distance in miles from downtown”. Similarly, we can use the independent variable \(t\) to represent the “number of hours after \(8\) 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.
If the variables represent actual things or events, we must define them to include the units we used to measure the event. This allows us to extract meaning from the values or expressions.
If you know Spanish you can interpret the meaning of a Spanish article you read by translating it into English. Likewise, if you know “math”, you can interpret the meaning of function notation by translating it into English, provided you know what the variables in the notation mean or what they are measuring.
If \(D\) represents distance, you must define the units in which the distance is measured: miles, inches, centimeters, light years, etc.
If \(t\) represents time, will it be measured with units of seconds, hours or years?
Since the position of the train depends on the amount of time that has passed, we say “the distance depends on time”. More specifically, we say that 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! The expression \(f(t)\) does not mean \(f\) times \(t\text{.}\)
The word “function” usually refers to the actual rule which turns the input into the output. Sometimes, however, we will use it to describe the output of the relationship.
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 \(S = f(t)\text{,}\) where \(f\) is the function, the instructions, that calculates your weekly salary. If you work \(12\) hours, then \(t = 12\text{,}\) 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. The expression \(f(12)\) literally means, “whatever you earn when you work \(12\) hours.”
In our train example we chose some input, a time, and the function gave us 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. In this case, the notation \(f(0)\) means “Whatever 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. For people, 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. For the train example we saw earlier, 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 \(t = 10\text{,}\) 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. Position 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 on its way out, then later in the afternoon on the way back into town. If we choose position 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.
A table of values is one way to express the inputs and outputs of a relation. In a vertical table, the inputs are on the left, and the outputs are on the right. In a horizontal table, the top row is the input, and the bottom row is the output.
Input | Output |
input | output |
input | output |
input | output |
Input | input | input | input |
Output | output | output | output |
It is easy to tell whether a graph represents a function or not. Remember that the inputs are found along the horizontal axis, and the outputs are found along the vertical axis. When deciding whether the graph is a function, 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: 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 function. If the vertical line crosses the graph anywhere at more than one point, then the graph is not a function.
It is 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 “use an input value in order to find an output value.” Therefore, evaluating \(f(2)\) 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 available.
The important thing is we need some input value (time) to determine the output value for the train's distance from downtown. Remember: 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 a placeholder for the actual input value, while “\(y\)” is a placeholder 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 \(x = 1\text{,}\) then we can evaluate \(f(1)\) by replacing the \(x\) in the formula with a \(1\text{.}\) Literally, \(f(1)\) means “what you get when you put \(1\) in the place of \(x\)”.
Similarly, if you change the input to \(x = 2\text{,}\) then you can evaluate the expression \(f(2)\text{,}\) which is “what you get when you put \(2\) in the place of \(x\)”.
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: \begin{equation*} f(\text{input})=\text{output} \end{equation*} Whatever is in the parenthesis is the input.
The input can be a simple number like \(2\) or \(-4\text{,}\) or it 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 that 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.
For the function \(f(x) = x^2 - x\text{,}\) you can put \(5\) into the formula and get the output: \begin{align*} f(5) \amp= 5^2 - 5\\ \amp= 20 \end{align*}
Or, you can put something complicated like \(\frac{x^2}{x^2 + 1}\) into the formula, 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 expression 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.
Solving means to work backward from a known output value, in order to determine all the inputs 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 give us that result. There may not be any, but there could be one, several or even infinite possible inputs that give \(y = 0\) 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 command:
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.” This equation may have zero, one, several, or even infinite 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.