Chapter 1: Functions & Models

1.1Four Ways to Represent a Function

Function

• Happens when one quantity depends on another

• Remember the function machine?

• Area of a circle is dependent on its radius• Human population increases with time• Cost of mailing a letter depends on the weight of

the letter

Definition

• A 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.

23456

-10123

Functions

• D and E are sets of real numbers

• Set D is the domain of the function

• f(x) is the “value of f at x” and reads “f of x”

• range of f is set of all values of f(x)

Functions

• Independent variable– A symbol that represents an arbitrary number in the

domain of a function f

• Dependent variable– A symbol that represents a number in the range of f

Function Machinex

f(x)

f(x) = …..

Arrow Diagram

x

a

f(x)

f(a)

Graph

• Most common way to visualize a function

• If f is a function with domain D, then its graph is the set of ordered pairs:

• **read as “the graph of f consists of all points (x,y) in the coordinate plane such that y=f(x) and x is in the domain of f”

Dxxfx )(,

Example 1• (a) Find the values

of f(1) and f(5)

• (b) What are the domain and range of f?

Example 2a• Sketch the graph and find the domain and range of

12)( xxf

Example 2b• Sketch the graph and find the domain and range of

2)( xxg

Example 3• If And

• evaluate

152)( 2 xxxf 0h

h

afhaf )()(

Difference Quotient

h

afhaf )()(

Represents the average rate of change of f(x) between x = a and x = a+h

Four Ways to Represent a Function

• 1. verbally (by a description in words)

• 2. numerically (by a table of values)

• 3. visually (by a graph)

• 4. algebraically (by an explicit formula)

Example 4• When you turn on a hot-water faucet, the

temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on.

Example 5• A rectangular storage container with an open top

has a volume of 10 m3. The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base.

Example 6a

• Find the domain of 2)( xxf

Example 6b

• Find the domain of xx

xg

2

1)(

Functions

• The graph of a function is a curve in the xy-plane

• Which curves in the xy-plane are graphs of functions?

Vertical Line Test

• Used with a graph of a function

• 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.

• Means that for each element in the domain of the function, there is only ONE element in the range

Is it a function?

Piecewise Defined Functions

• Defined by different formulas in different parts of their domains

Example 7• A function f is defined by

• Evaluate f(0), f(1), and f(2) and sketch the graph

1,

1,1)(

2 xx

xxxf

Absolute Value

• |a| = a if a ≥ 0

• |a| = -a if a < 0

• Remember if a is negative, then –a is positive!

Example 8

• Sketch the graph of xxf )(

Example 9

• Find a formula for the function 9 (figure 17)

Example 10• In Example C at the beginning of this section in the

book, we considered the cost C(w) of mailing a first-class letter with weight w. In effect, this is a piecewise defined function because, from the table of values, we have:

• Called a step function

...

43,11.1

32,87.0

21,63.0

10,39.0

)(

w

w

w

w

wC

Even Function

• If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is an even function

• These are symmetric functions with respect to the y-axis

Odd Functions

• If f satisfies f(-x) = -f(x) for every number x in its domain, then f is called an odd function

• These are symmetric about the origin (or rotated 180 degrees)

Example 11a

• Determine if the function is even, odd, or neither

xxxf 5)(

Example 11b

• Determine if the function is even, odd, or neither41)( xxg

Example 11c

• Determine if the function is even, odd, or neither22)( xxxh

Increasing vs Decreasing

• Increasing if f(x1) < f(x2) whenever x1 < x2

• Decreasing if f(x1) > f(x2) whenever x1 < x2

Homework

• P. 20

• 1, 5-8, 11, 13, 23-43 odd, 65, 67