Four Ways to Represent a Function

A function is a relation between a set of inputs and a set of permissible outputs, provided that each input is related to exactly one output. An example is the function that relates each real number $x$ to its square $x^2$. The output of a function $f$ corresponding to an input $x$ is denoted by $f(x)$ (read "f of x"). In this example, if the input is $-3$, then the output is 9, and we may write $f(-3)=9$. The input variable(s) are sometimes referred to as the argument(s) of the function.

Modern calculus texts emphasize that a function can be expressed in four different ways.

Verbal: When modeling a process mathematically, one often first develops a verbal description of the problem. For example, the expression $2x+6$ can be represented as "Double x and add six" or "six added to two times x".

Algebraic: This is the most common, most concise, and most powerful representation: $2x+6$. Note that in an algebraic representation, the input number is represented as a variable (in this case, an x).

Numerical: This can be expressed as a list of value pairs, as in $(4,14)$ — meaning that if a 4 goes in, a 14 comes out. (You may recognize this as the $(x,y)$ points used in graphing.)

Graphical: This involves modeling a function in a dimensional overlay. Scientific data is often recorded in a visual format. Examples include seismograph readings, electrocardiograms, and oscilloscope readings.

These are not four different types of functions; they are four different views of the same function. One of the most important skills in algebra and calculus is being able to convert a function between these different forms, and this theme will recur in different forms throughout the text.