1.11.11.11.1

FunctionsFunctions

Quick Review 2

2

2

2

4 2

Factor the following expressions completely over the real numbers.

1. 9

2. 4 81

3. 8 16

4. 2 7 3

5. 3 4

x

y

x x

x x

x x

What you’ll learn about• Numeric Models• Algebraic Models• Graphic Models• The Zero Factor Property• Problem Solving• Grapher Failure and Hidden Behavior• A Word About Proof

… and whyNumerical, algebraic, and graphical models provide

differentmethods to visualize, analyze, and understand data.

Mathematical ModelA mathematical model is a mathematical

structure that approximates phenomena for the

purpose of studying or predicting their behavior.

Numeric ModelA numeric model is a kind of mathematicalmodel in which numbers (or data) are analyzed to gain insights into phenomena.

Algebraic ModelAn algebraic model uses formulas to relatevariable quantities associated with thephenomena being studied.

Example Comparing Pizzas

A pizzeria sells a rectangular 20" by 22" pizza for the same price as its

large round pizza (24" diameter). If both pizzas are the same thickness,

which option gives the most pizza for the money?

Example Comparing Pizzas

A pizzeria sells a rectangular 20" by 22" pizza for the same price as its

large round pizza (24" diameter). If both pizzas are the same thickness,

which option gives the most pizza for the money?

2

2

Compare the areas of the pizzas.

Rectangular pizza: Area 20 22 440 square inches

24Circular pizza: Area 144 452.4 square inches

2

The round pizza is larger and therefore gives more for t

l w

r

he money.

Graphical ModelA graphical model is a visible

representation of a numerical model or an algebraic model thatgives insight into the relationships betweenvariable quantities.

Example Solving an Equation

2Solve the equation 8 4 algebraically.x x

Example Solving an Equation

2Solve the equation 8 4 algebraically.x x

2

Set the given equation equal to zero:

4 8 0

Use the quadratic formula to solve for .

4 16 32

2

4 48

2

4 4 3

2

2 2 3

Approximations for the solutions are 1.4641 and -5.4641.

x x

x

x

x x

Fundamental Connection

If is a real number that solves the equation ( ) 0, then these three

statements are equivalent:

1. The number is a root (or solution) of the equation ( ) 0.

2. The number is a zero of ( ).

3. T

a f x

a f x

a y f x

he number is an -intercept of the graph of ( ).a x y f x

Pólya’s Four Problem-Solving Steps

1. Understand the problem.2. Devise a plan.3. Carry out the plan.4. Look back.

A Problem-Solving Process

Step 1 – Understand the problem.• Read the problem as stated, several

times if necessary.• Be sure you understand the meaning of

each term used.• Restate the problem in your own words.

Discuss the problem with others if you can.

• Identify clearly the information that you need to solve the problem.

• Find the information you need from the given data.

A Problem-Solving Process

Step 2 – Develop a mathematical model of the problem.

• Draw a picture to visualize the problem situation. It usually helps.

• Introduce a variable to represent the quantity you seek.

• Use the statement of the problem to find an equation or inequality that relates the variables you seek to quantities that you know.

A Problem-Solving Process

Step 3 – Solve the mathematical model and support or confirm the solution.

• Solve algebraically using traditional algebraic models and support graphically or support numerically using a graphing utility.

• Solve graphically or numerically using a graphing utility and confirm algebraically using traditional algebraic methods.

• Solve graphically or numerically because there is no other way possible.

A Problem-Solving Process

Step 4 – Interpret the solution in the problem setting.

• Translate your mathematical result into the problem setting and decide whether the result makes sense.

Example Seeing Grapher Failure

Look at the graph of 3 /(2 5) on a graphing calculator.

Is there an -intercept?

y x

x

Example Seeing Grapher Failure

Look at the graph of 3 /(2 5) on a graphing calculator.

Is there an -intercept?

y x

x

The graph is shown below. Notice that the graph appears to show an

-intercept between 2 and 3. Confirm this algebraically:

30

2 50 2 5 3

0 3 This statement is false for all , so there is no -inte

x

xx

x x

rcept.

The grapher plots points at regular increments from left to right,

connecting the points as it goes.

1.1(a)/1.21.1(a)/1.21.1(a)/1.21.1(a)/1.2

Functions and Their PropertiesFunctions and Their Properties

Quick Review

2

2

Solve the equation or inequality.

1. 9 0

2. 16 0

Find all values of algebraically for which the algebraic

expression is not defined.

13.

3

4. 3

15.

3

x

x

x

x

x

x

x

What you’ll learn about• Function Definition and Notation• Domain and Range• Continuity• Increasing and Decreasing Functions• Boundedness• Local and Absolute Extrema• Symmetry• Asymptotes• End Behavior

… and whyFunctions and graphs form the basis for understanding The mathematics and applications you will see both in your

work place and in coursework in college.

Function, Domain, and Range

A function from a set D to a set R is a rule that

assigns to every element in D a unique element

in R. The set D of all input values is the domain

of the function, and the set R of all output values

is the range of the function.

Mapping

Example Seeing a Function Graphically

Of the three graphs shown below, which is not the graph of a function?

Example Seeing a Function Graphically

Of the three graphs shown below, which is not the graph of a function?

The graph in (c) is not the graph of a function. There are three points on the graph with x-coordinates 0.

Vertical Line TestA graph (set of points (x,y)) in the

xy-planedefines y as a function of x if and

only if no vertical line intersects the graph in

more than onepoint.

Agreement

Unless we are dealing with a model that

necessitates a restricted domain, we will assume that the domain of a function defined by an algebraic expression is the same as the domain of the algebraic expression, the implied domain.

For models, we will use a domain that fits the situation, the relevant domain.

Example Finding the Domain of a Function

Find the domain of the function.

( ) 2f x x

Example Finding the Domain of a

Function

Find the domain of the function.

( ) 2f x x

Solve algebraically:

The expression under a radical may not be negative.

2 0

2

The domain of is the interval [ 2, ).

x

x

f

Example Finding the Range of a Function

2Find the range of the function ( ) .f x

x

Example Finding the Range of a Function

2

Find the range of the function ( ) .f xx

Solve Graphically:

2The graph of shows that the range is all real numbers except 0.

The range in interval notation is ,0 0, .

yx

Continuity

Example Identifying Points of Discontinuity

Which of the following figures shows functions that are

discontinuous at x = 2?

Example Identifying Points of Discontinuity

Which of the following figures shows functions that are

discontinuous at x = 2?

The function on the right is not defined at x = 2 and can not be continuous there. This is a removable discontinuity.

Increasing and Decreasing Functions

Increasing, Decreasing, and Constant Function

on an IntervalA function f is increasing on an interval if, for

any two points in the interval, a positive change in x results in a positive change in f(x).

A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x).

A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x).

Example Analyzing a Function for

Increasing-Decreasing Behavior2

2Given ( ) . Tell the intervals on which ( ) is increasing and the

1intervals on which it is decreasing.

xg x g x

x

Example Analyzing a Function for

Increasing-Decreasing Behavior

2

2Given ( ) . Tell the intervals on which ( ) is increasing and the

1intervals on which it is decreasing.

xg x g x

x

From the graph, we see that ( ) is increasing on , 1 , increasing on

( 1,0], decreasing on [0,1), and decreasing on (1, ).

g x

Lower Bound, Upper Bound and Bounded

A function f is bounded below of there is somenumber b that is less than or equal to everynumber in the range of f. Any such number b iscalled a lower bound of f.

A function f is bounded above of there is somenumber B that is greater than or equal to everynumber in the range of f. Any such number B iscalled a upper bound of f.

A function f is bounded if it is bounded both above and below.

Local and Absolute Extrema

A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f.

A local minimum of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f.

Local extrema are also called relative extrema.

Example Identifying Local Extrema

4 2Find the local maxima or local minima of ( ) 7 6 . Find the

values of where each local maximum and local minimum occurs.

f x x x x

x

Example Identifying Local Extrema

4 2Find the local maxima or local minima of ( ) 7 6 . Find the

values of where each local maximum and local minimum occurs.

f x x x x

x

The graph of the function suggests that there are two local minimum value and

one local maximum value. Use the calculator to approximate local minima as

-24.06 (which occurs at -2.06) and -1.77 (whicx h occurs at 1.60). The

local maximum is 1.32 (which occurs at 0.46).

x

x

Symmetry with respect to the y-axis

Symmetry with respect to the x-axis

Symmetry with respect to the origin

Example Checking Functions for

Symmetry2

Tell whether the following function is odd, even, or neither.

( ) 3f x x

Example Checking Functions for

Symmetry 2

Tell whether the following function is odd, even, or neither.

( ) 3f x x

2

2

Solve Algebraically:

Find (- ).

(- ) (- ) 3

3

( ) The function is even.

f x

f x x

x

f x

Horizontal and Vertical Asymptotes

The line is a horizontal asymptote of the graph of a function ( )

if ( ) approaches a limit of as approaches + or - .

In limit notation: lim ( ) or lim ( ) .

The line is a ver

x x

y b y f x

f x b x

f x b f x b

x a

tical asymptote of the graph of a function ( )

if ( ) approaches a limit of + or - as approaches from either

direction.

In limit notation: lim ( ) or lim ( ) .x a x a

y f x

f x x a

f x f x