Chapter 15
FUNCTIONS
Though a fundamental concept of mathematics, the function is little more
than an expression that meets a specific criterion—each independent
variable input results in exactly one dependent variable output. Cosmeti-
cally, the function differs from a general expression because it is assigned
a name, often f(x). This chapter investigates the distinction that makes a
mathematical relation a function, operations performed on functions, the
composition of functions, inverse functions, and piecewise-defined func-
tions.
Many books describe a function as a machine: you plug in an x-value,
and you get out an f(x)-value. That description works, but it might be
easier to think of a function as an expression you can refer to by name.
For instance, let’s say f(x) = x
2
+ 1. After you dene it, every time you
refer to f(x), everyone knows you’re talking about x
2
+ 1.
There’s other benets as well. Writing f(2) means “plug x = 2 into the
function f(x).” So if f(x) = x
2
+ 1, then f(2) = 2
2
+ 1 = 5.
Functions are more than just named expressions, however. A function has
to meet one requirement—every x-value you plug into it has to produce
exactly one output. In other words, if you plug x = 1 into a function, you’ll
never get two different answers.
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