Section 1-3Section 1-3

The Twelve Basic FunctionsThe Twelve Basic Functions

Section 1-3Section 1-3

• the twelve basic functionsthe twelve basic functions

• studying their domain, continuity, studying their domain, continuity, boundedness, and symmetryboundedness, and symmetry

• analyzing functions graphicallyanalyzing functions graphically

• piecewise functionspiecewise functions

The Twelve Basic FunctionsThe Twelve Basic Functions the 12 functions are found on page 102-103the 12 functions are found on page 102-103 study them carefully so you know the study them carefully so you know the

characteristics of each functioncharacteristics of each function some of these functions you have already some of these functions you have already

studied studied and know well (identity, squaring, and know well (identity, squaring, cubing, cubing, absolute value)absolute value)

some of these functions you have already some of these functions you have already studied studied but we will cover them again but we will cover them again (reciprocal, (reciprocal, square root, exponential, log., square root, exponential, log., sine, cosine)sine, cosine)

some of these functions are new and will be some of these functions are new and will be also also covered this year (greatest int. and covered this year (greatest int. and logistic)logistic)

DomainDomain

Question: Nine of the functions have a Question: Nine of the functions have a domain of all reals, which three do domain of all reals, which three do not?not?

Answer: square root, natural log., and Answer: square root, natural log., and reciprocalreciprocal

ContinuityContinuity

Question: Which two functions have Question: Which two functions have points of discontinuity?points of discontinuity?

Answer: reciprocal and greatest Answer: reciprocal and greatest integerinteger

BoundednessBoundedness

Question: Which of the three functions Question: Which of the three functions are bounded (above and below)?are bounded (above and below)?

Answer: sine, cosine, and logisticAnswer: sine, cosine, and logistic

SymmetrySymmetry

Question: Which of the functions are Question: Which of the functions are even?even?

Answer: squaring, cosine, absolute Answer: squaring, cosine, absolute valuevalue

Analyzing Functions Analyzing Functions GraphicallyGraphically• some functions we study will be an some functions we study will be an

alteration of one of the basic 12 alteration of one of the basic 12 • if we are asked to analyze such a if we are asked to analyze such a

function we will need to graph it (by function we will need to graph it (by hand or with the graphing calculator)hand or with the graphing calculator)

• look at each characteristic of look at each characteristic of functions that we studied in the last functions that we studied in the last section in respect to the graph section in respect to the graph (domain, range, symmetry, local (domain, range, symmetry, local extrema, incr/decr, continuity, extrema, incr/decr, continuity, boundedness, asymptotes, and end boundedness, asymptotes, and end behavior)behavior)

Piecewise-Defined FunctionsPiecewise-Defined Functions• these are functions that come in these are functions that come in

parts, having a different equation for parts, having a different equation for different values of the domaindifferent values of the domain

• many piecewise functions have jump many piecewise functions have jump discontinuitiesdiscontinuities

• here is an example:here is an example:

2

2 if 0( )

if 0

x xf x

x x