The Concept of a Function

A quick review of useful functions and definitions introduced throughout precalculus, that will make many appearances during our course.

What is a Function?

A function is a precise, unambiguous rule for assigning each element of one set (called the domain) to a unique element of another set (called the range). Schematically, you’ll often see a function represented like this, to emphasize the three important parts of its definition:

A function may be given by any such rule, between any two sets (for example, the rule which assigns each human being on earth to the date and time of their birth is a function!) but we will concentrate our interests primarily on functions for which both the domain and the range are subsets of the real number line. These ‘real valued’ functions are important across the STEM disciplines as a means of quantifying phenomena, and also comprise the rich mathematical playground in which calculus was first discovered.

The graph of a function is the set of ordered pairs $(x, f (x))$ for all elements $x$ of the domain: when both the domain and range are subsets of the number line $R$ , the graph can be visualized as a curve in the plane.

Drawing graphs is a great way to get a qualitative understanding of the behavior of a function. Things like what the function is doing (is it increasing in value, or decreasing?) does it reach a maximum value and then ’turn around’? Etc. Indeed - we will discover much of calculus while trying to quantify some of these very notions!

1-to-1 and Onto

For every function, the output is unique for a given input, but it is absolutely allowable that many different inputs may give the same output (returning to the birthday example, any single person cannot have two different birthdays, but its relatively common for two distinct people to share the same birthdate!). When each output does come from a unique input, we call the function one-to-one (meaning every one input goes to one output, with no overlaps).

A function is also not required to have every single element of the range actually ‘hit’ by the output of some element of the domain. When this does happen, the function is called onto.

Function Composition

Some of the real power of working with functions comes from the ease of being able to create complex combinations, describing widely varying phenomena, out of very simple pieces. The most important tool in doing so is composition, or creating a new function by stringing two existing functions together. Pictorally, a composition looks like this:

PICTURE

The composition of $g$ , followed by $f$ is written as $f \circ g$ , or $f (g (x))$ , and is only defined when the outputs of $g$ are possible inputs of $f$ . (For example, if $g$ is a function taking people to their birthdates, and $f$ is the function taking books to their current price on Amazon, the compostion $f \circ g$ makes no sense!)

Inverse Functions

If a function is a rule for translating points of one set into another, the inverse function is a way of undoing this, and returning the outputs to the inputs they came from.

PICTURE

If $f : A \to B$ is a function and $g : B \to A$ is its inverse, the condition that $g$ puts everything back where it came from before $f$ is translated into symbols as $g (f (x)) = x$ , for every $x$ in the domain $A$ . It turns out as a consequence, that whenever this is true $f (g (x)) = x$ as well for all $x$ in $B$ (so, $f$ undoes $g$ as well!)

A function is called invertible if it has an inverse. Note that not every function is invertible! As a classic example, consider the function $f (x) = 0$ - if every number is sent to $0$ , then the inverse should send $0$ to every number! But this cannot be a function, as a function needs to have a unique output for every inuput. When a function $f$ does have an inverse, it is often denoted by the symbol $f^{- 1}$ , except in the case of exponentials where the inverse of $a^{x}$ is $\log_{a} (x)$ , and trigonometric functions, where the inverse of $\sin, \cos$ are written either $\sin^{- 1}, \cos^{- 1}$ or $\arcsin, \arccos$ .