Functions

A function is a way of assigning one set of numbers to another set of numbers. For example, let's say that I am at my home and I want to walk through downtown and across my city. As I do this I keep track of how far I am from home and I also measure the height of the buildings as I go. (Let's say I measure the height of the buildings on my right-hand side.) So at every distance from home there is a height of a building. This is a function. My function may look something like the following:

In general we label the horizontal axis the "x-axis", the vertical axis the "y-axis", and the function as "f(x)". So a typical function would look something like:

In the real world we would describe a function as we did above with the height of the buildings. However, in Calculus we speak about functions in a way that allows us to do more advanced mathematics on them. A more rigorous definition of a function will include four main elements of the function:

• A rule relating the x-axis with the y-axis. Typically this is an algebraic, trigonometric or other relationship such as $x^2$, <Sign in to see all the formulas>, <Sign in to see all the formulas>, or <Sign in to see all the formulas>. For example, we would denote the function $x^2$ as $y=x^2$ or as <Sign in to see all the formulas>.
• For every value of $x$ there is only one value of $y$. (We can't have two buildings on the same spot!) So while $y=x^2$ is a function, <Sign in to see all the formulas> is not a function. Let's say that $x=4$ then for $y=x^2$, $y=16$, but for <Sign in to see all the formulas>, $y$ can take on the value of both $+2$ and $-2$. Now if we said <Sign in to see all the formulas> or <Sign in to see all the formulas> then these would be functions and we could denote them by <Sign in to see all the formulas> or <Sign in to see all the formulas>. (For <Sign in to see all the formulas> this fact is so well known that we usually see <Sign in to see all the formulas> which should always be taken to mean <Sign in to see all the formulas> and <Sign in to see all the formulas> to mean <Sign in to see all the formulas>.)
• The domain of the function is the portion of the x-axis for which the function is defined. (By defined we mean that $f(x)$ is a number that we know.) Whenever the domain is not explicitly specified we assume that the domain is all real numbers for which $f(x)$ is a real number. So for <Sign in to see all the formulas> then the domain is all real numbers. However, for <Sign in to see all the formulas> the domain is all positive numbers because the square root of a negative number is not a real number. When we do need to explicitly define the domain of a function we will do it with a bracket that explicitly dictates what rule relating the $x$ and $y$ axes is used for which domain. For example,
  

  <Sign in to see all the formulas>

  
  denotes a function whose domain is not made up of all real numbers. The domain is <Sign in to see all the formulas> which is a union of open and closed intervals.
• The range of the function is the portion of the y-axis for which the function is defined. The range is determined from knowing the domain and the function $f(x)$.

Here are some examples where we determine the domain, range, and graph of a function:

Function	Domain	Range	Graph
<Sign in to see all the formulas>	All real numbers: <Sign in to see all the formulas>	All positive real numbers: <Sign in to see all the formulas>	Enlarge
<Sign in to see all the formulas>	All real numbers: <Sign in to see all the formulas>	All real numbers: <Sign in to see all the formulas>	Enlarge
<Sign in to see all the formulas>	All positive real numbers: <Sign in to see all the formulas>	All negative real numbers: <Sign in to see all the formulas>	Enlarge
<Sign in to see all the formulas>	All real numbers: <Sign in to see all the formulas>	All real numbers between -1 and 1: <Sign in to see all the formulas>	Enlarge
<Sign in to see all the formulas>	<Sign in to see all the formulas>	<Sign in to see all the formulas>	Enlarge

Now try graphing some of your own functions on our Function Wizard!