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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

variable.

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

than once.

Functions that are defined by different formulas in different parts of

their domains are called piecewise defined functions.

Symmetry:

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.

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Increasing and decreasing functions:

oA function f is called increasing on an interval I if

oIt is called decreasing on I if

1.2 Mathematical Models: A catalog of essential functions

Linear Models

owhere m is the slope and b is the y-intercept.

Polynomials

oA function is called a polynomial if

where n is a

nonnegative integer and the numbers are

constants called the coefficients of the polynomial.

oThe degree of the polynomial is n.

oA polynomial of degree 2 is called a quadratic function.

oA polynomial of degree 3 is called a cubic function.

Power Functions

o

o

o

oIt’s called a root function.

o

oIt’s called a reciprocal function.

Rational Functions

oA rational function f is a ratio of two polynomials:

Algebraic Functions

oA function is called an algebraic function if it can be

constructed using algebraic operations (such as addition,

subtraction, multiplication, division, and taking roots) starting

with polynomials.

Trigonometric Functions

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