Functions and Models
1.1 Four ways to represent a function
Representations of functions
oA function f is a rule that assigns to each element x in a set D
exactly one element, called f(x), in a set E.
oThe set D is called the domain of the function.
oThe number f(x) is the value of f at x and is read “f of x.”
oThe range of f is the set of all possible values of f(x) as x
varies throughout the domain.
oA symbol that represents an arbitrary number in the domain of
a function f is called an independent variable. A symbol that
represents a number in the range of f is called a dependent
oAnother way to picture a function is by an arrow diagram.
oThe most common method for visualizing a function is its
graph. If f is a function with domain D, then its graph is the set
of ordered pairs .
oThere are four possible ways to represent a function:
Verbally (by a description in words)
Numerically (by a table of values)
Visually (by a graph)
Algebraically (by an explicit formula)
The Vertical Line Test A curve in the xy-plane is the graph of a
function of x if and only if no vertical line intersects the curve more
Functions that are defined by different formulas in different parts of
their domains are called piecewise defined functions.
oIf a function f satisfies f(-x)=f(x) for every number x in its
domain, then f is called an even function. The geometric
significance of an even function is that its graph is symmetric
with respect to the y-axis.
oIf f satisfies f(-x)=-f(x) for every number x in its domain, then is
called an odd function. The graph of an odd function is
symmetric about the origin.