Transcript Slide 1

Functions and
Their Graphs
Chapter 2
Functions
Section 2.1
Relations

Relation: A correspondence between
two sets.

x corresponds to y or y depends on x
if a relation exists between x and y

Denote by x ! y in this case.
Relations

Example.
Person
Melissa
John
Jennifer
Patrick
Salary
$45,000
$40,000
$50,000
Relations

Example.
Number
Number
0
0
1
1
{1
4
2
{2
Functions

Function: special kind of relation
Each input corresponds to precisely one
output
 If X and Y are nonempty sets, a
function from X into Y is a relation
that associates with each element of X
exactly one element of Y

Functions

Example.
Problem: Does this relation represent a function?
Answer:
Person
Melissa
John
Jennifer
Patrick
Salary
$45,000
$40,000
$50,000
Functions

Example.
Problem: Does this relation represent a function?
Answer:
Number
Number
0
0
1
1
{1
4
2
{2
Domain and Range

Function from X to Y
Domain of the function: the set X.
 If x in X:



The image of x or the value of the function
at x: The element y corresponding to x
Range of the function: the set of all
values of the function
Domain and Range

Example.
Problem: What is the range of this function?
Answer:
X
Y
{3
0
1
4
9
{2
{1
0
1
2
3
Domain and Range

Example. Determine whether the
relation represents a function. If it is
a function, state the domain and
range.
Problem:
Relation: f(2,5), (6,3), (8,2), (4,3)g
Answer:
Domain and Range

Example. Determine whether the
relation represents a function. If it is
a function, state the domain and
range.
Problem:
Relation: f(1,7), (0, {3), (2,4), (1,8)g
Answer:
Equations as Functions

To determine whether an equation is
a function

Solve the equation for y.
If any value of x in the domain corresponds
to more than one y, the equation doesn’t
define a function
 Otherwise, it does define a function.

Equations as Functions

Example.
Problem: Determine if the equation
x + y2 = 9
defines y as a function of x.
Answer:
Function as a Machine


Accepts numbers from domain as
input.
Exactly one output for each input.
Finding Values of a
Function

Example. Evaluate each of the following for the
function
f(x) = {3x2 + 2x
(a) Problem: f(3)
Answer:
(b) Problem: f(x) + f(3)
Answer:
(c) Problem: f({x)
Answer:
(d) Problem: {f(x)
Answer:
(e) Problem: f(x+3)
Answer:
Finding Values of a
Function

Example. Evaluate the difference
quotient
of the function
Problem: f(x) = { 3x2 + 2x.
Answer:
Implicit Form of a Function
A function given in terms of x
and y is given implicitly.
 If we can solve an equation for y
in terms of x, the function is
given explicitly

Implicit Form of a Function

Example. Find the explicit form of
the implicit function.
(a) Problem: 3x + y = 5
Answer:
(b) Problem: xy + x = 1
Answer:
Important Facts




For each x in the domain of f, there
is exactly one image f(x) in the range
An element in the range can result
from more than one x in the domain
We usually call x the independent
variable
y is the dependent variable
Finding the Domain

If the domain isn’t specified, it will
always be the largest set of real
numbers for which f(x) is a real
number

We can’t take square roots of negative
numbers (yet) or divide by zero
Finding the Domain

Example. Find the domain of
each of the following functions.
(a) Problem: f(x) = x2 { 9
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Finding the Domain

Example. A rectangular garden has a
perimeter of 100 feet.
(a) Problem: Express the area A of the
garden as a function of the width w.
Answer:
(b) Problem: Find the domain of A(w)
Answer:
Operations on Functions

Arithmetic on functions f and g

Sum of functions:
(f + g)(x) = f(x) + g(x)

Difference of functions:
(f { g)(x) = f(x) { g(x)

Domains: Set of all real numbers in the
domains of both f and g.

For both sum and difference
Operations on Functions

Arithmetic on functions f and g

Product of functions f and g is
(f ¢ g)(x) = f(x) ¢ g(x)

The quotient of functions f and g is
f 
f ( x)
 ( x) 
g( x)
 g


Domain of product: Set of all real numbers in
the domains of both f and g
Domain of quotient: Set of all real numbers in
the domains of both f and g with g(x)  0
Operations on Functions

Example. Given f(x) = 2x2 + 3 and
g(x) = 4x3 + 1.
(a) Problem: Find f+g and its domain
Answer:
(b) Problem: Find f { g and its domain
Answer:
Operations on Functions

Example. Given f(x) = 2x2 + 3 and
g(x) = 4x3 + 1.
(c) Problem: Find f¢g and its domain
Answer:
(d) Problem: Find f/g and its domain
Answer:
Key Points
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Relations
Functions
Domain and Range
Equations as Functions
Function as a Machine
Finding Values of a Function
Implicit Form of a Function
Important Facts
Finding the Domain
Key Points (cont.)

Operations on Functions
The Graph of a
Function
Section 2.2
Vertical-line Test

Theorem. [Vertical-Line Test]
A set of points in the xy-plane is the
graph of a function if and only if
every vertical line intersects the
graphs in at most one point.
Vertical-line Test

Example.
Problem: Is the graph that of a function?
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Vertical-line Test

Example.
Problem: Is the graph that of a function?
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Finding Information From
Graphs

Example. Answer the
questions about the graph.
0,4
4
(a) Problem: Find f(0)
Answer:
1,2
2
1,2
4
(b) Problem: Find f(2)
4
2,
2,
5
5
Answer:
-4
-2
2
(c) Problem: Find the domain
Answer:
-2
(d) Problem: Find the range
-4
Answer:
4
Finding Information From
Graphs

Example. Answer the
questions about the graph.
0,4
4
(e) Problem: Find the
x-intercepts:
1,2
Answer:
(f) Problem: Find the
y-intercepts:
Answer:
2
1,2
4
4
2,
2,
5
-4
5
-2
2
-2
-4
4
Finding Information From
Graphs

Example. Answer the
questions about the graph.
0,4
4
(g) Problem: How often does
the line y = 3 intersect the
graph?
1,2
Answer:
(i) Problem: For what values
of x is f(x) > 0?
Answer:
1,2
4
Answer:
(h) Problem: For what values
of x does f(x) = 2?
2
4
2,
2,
5
-4
5
-2
2
-2
-4
4
Finding Information From
Formulas

Example. Answer the following questions
for the function
f(x) = 2x2 { 5
(a) Problem: Is the point (2,3) on the graph of
y = f(x)?
Answer:
(b) Problem: If x = {1, what is f(x)? What is the
corresponding point on the graph?
Answer:
(c) Problem: If f(x) = 1, what is x? What is (are)
the corresponding point(s) on the graph?
Answer:
Key Points
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Vertical-line Test
Finding Information From Graphs
Finding Information From Formulas
Properties of
Functions
Section 2.3
Even and Odd Functions

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Even function:

For every number x in its domain, the
number {x is also in the domain

f({x) = f(x)
Odd function:

For every number x in its domain, the
number {x is also in the domain

f({x) = {f(x)
Description of Even and
Odd Functions

Even functions:


If (x, y) is on the graph, so is ({x, y)
Odd functions:

If (x, y) is on the graph, so is ({x, {y)
Description of Even and
Odd Functions

Theorem.
A function is even if and only if its
graph is symmetric with respect to
the y-axis.
A function is odd if and only if its
graph is symmetric with respect to
the origin.
Description of Even and Odd
Functions

Example.
Problem: Does
the graph
represent a
function which
is even, odd, or
neither?
Answer:
4
2
-4
-2
2
-2
-4
4
Description of Even and Odd
Functions

Example.
Problem: Does
the graph
represent a
function which
is even, odd, or
neither?
Answer:
4
2
-4
-2
2
-2
-4
4
Description of Even and Odd
Functions

Example.
Problem: Does
the graph
represent a
function which
is even, odd, or
neither?
Answer:
4
2
-4
-2
2
-2
-4
4
Identifying Even and Odd
Functions from the Equation

Example. Determine whether the
following functions are even, odd or
neither.
(a) Problem:
Answer:
(b) Problem: g(x) = 3x2 { 4
Answer:
(c) Problem:
Answer:
Increasing, Decreasing and
Constant Functions

Increasing function (on an open interval
I):


Decreasing function (on an open interval
I)


For any choice of x1 and x2 in I, with
x1 < x2, we have f(x1) < f(x2)
For any choice of x1 and x2 in I, with
x1 < x2, we have f(x1) > f(x2)
Constant function (on an open interval I)

For all choices of x in I, the values f(x) are
equal.
Increasing, Decreasing and
Constant Functions
Increasing, Decreasing and
Constant Functions

Example. Answer the
questions about the
function shown.
(a) Problem: Where is the
function increasing?
Answer:
(b) Problem: Where is the
function decreasing?
Answer:
(c) Problem: Where is the
function constant
Answer:
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Increasing, Decreasing and
Constant Functions
WARNING!


Describe the behavior
of a graph in terms of
its x-values.
Answers for these
questions should be
open intervals.
6
4
2
-6
-4
-2
2
-2
-4
-6
4
6
Local Extrema

Local maximum at c:



Local minimum at c:



Open interval I containing x so that, for all
x · c in I, f(x) · f(c).
f(c) is a local maximum of f.
Open interval I containing x so that, for all
x · c in I, f(x) ¸ f(c).
f(c) is a local minimum of f.
Local extrema:

Collection of local maxima and minima
Local Extrema

For local maxima:
Graph is increasing to the left of c
 Graph is decreasing to the right of c.


For local minima:
Graph is decreasing to the left of c
 Graph is increasing to the right of c.

Local Extrema

Example. Answer the
questions about the
given graph of f.
6
(a) Problem: At which
4
number(s) does f have a
2
local maximum?
Answer:
-7.5
-5
-2.5
2.5
-2
(b) Problem: At which
-4
number(s) does f have a
local minimum?
Answer:
-6
5
7.5
Average Rate of Change

Slope of a line can be interpreted as
the average rate of change

Average rate of change: If c is in the
domain of y = f(x)

Also called the difference quotient of f
at c
Average Rate of Change

Example. Find the average rates of
change of
(a) Problem: From 0 to 1.
Answer:
(b) Problem: From 0 to 3.
Answer:
(c) Problem: From 1 to 3:
Answer:
Secant Lines

Geometric interpretation to the
average rate of change
Label two points (c, f(c)) and (x, f(x))
 Draw a line containing the points.
 This is the secant line.


Theorem. [Slope of the Secant Line]
The average rate of change of a
function equals the slope of the secant
line containing two points on its
graph
Secant Lines
Secant Lines

Example.
15
Problem: Find an
12.5
equation of the
10
secant line to
7.5
5
containing (0, f(0))
and (5, f(5))
Answer:
2.5
-7.5
-5
-2.5
2.5
-2.5
-5
5
7.5
Key Points
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Even and Odd Functions
Description of Even and Odd
Functions
Identifying Even and Odd Functions
from the Equation
Increasing, Decreasing and Constant
Functions
Local Extrema
Average Rate of Change
Key Points (cont.)

Secant Lines
Linear Functions
and Models
Section 2.4
Linear Functions

Linear function:



Function of the form f(x) = mx + b
Graph: Line with slope m and y-intercept b.
Theorem. [Average Rate of Change of
Linear Function]
Linear functions have a constant average
rate of change. The constant average rate
of change of f(x) = mx + b is
Linear Functions

Example.
10
Problem: Graph the
linear function
f(x) = 2x { 5
Answer:
7.5
5
2.5
-10
-5
5
-2.5
-5
-7.5
-10
10
Application: Straight-Line
Depreciation

Example. Suppose that a company
has just purchased a new machine for
its manufacturing facility for
$120,000. The company chooses to
depreciate the machine using the
straight-line method over 10 years.
For straight-line depreciation, the
value of the asset declines by a fixed
amount every year.
Application: Straight-Line
Depreciation

Example. (cont.)
(a) Problem: Write a linear function that
expresses the book value of the machine
as a function of its age, x
Answer:
(b) Problem: Graph the linear function
Answer:
140000
120000
100000
80000
60000
40000
20000
-20000
-40000
2
4
6
8
10
12
14
Application: Straight-Line
Depreciation

Example. (cont.)
(c) Problem: What is the book value of
the machine after 4 years?
Answer:
(d) Problem: When will the machine be
worth $20,000?
Answer:
Scatter Diagrams

Example. The amount of money that
a lending institution will allow you to
borrow mainly depends on the
interest rate and your annual income.
The following data represent the
annual income, I, required by a bank
in order to lend L dollars at an
interest rate of 7.5% for 30 years.
Scatter Diagrams

Example. (cont.)
Annual Income,
I ($)
Loan Amount,
L ($)
15,000
44,600
20,000
59,500
25,000
74,500
30,000
89,400
35,000
104,300
40,000
119,200
45,000
134,100
50,000
149,000
55,000
163,900
60,000
178,800
65,000
193,700
70,000
208,600
Scatter Diagrams

Example. (cont.)
Problem: Use a graphing utility to draw a
scatter diagram of the data.
Answer:
Linear and Nonlinear
Relationships
Linear
Nonlinear
Nonlinear
Linear
Linear
Nonlinear
Line of Best Fit

For linearly related scatter diagram
Line is line of best fit.
 Use graphing calculator to find


Example.
(a) Problem: Use a graphing utility to find
the line of best fit to the data in the last
example.
Answer:
Line of Best Fit

Example. (cont.)
(b) Problem: Graph the line of best fit
from the last example on the scatter
diagram.
Answer:
Line of Best Fit

Example. (cont.)
(c) Problem: Determine the loan amount
that an individual would qualify for if
her income is $42,000.
Answer:
Direct Variation


Variation or proportionality.
y varies directly with x, or is
directly proportional to x:


There is a nonzero number such that
y = kx.
k is the constant of proportionality.
Direct Variation

Example. Suppose y varies directly
with x. Suppose as well that y = 15
when x = 3.
(a) Problem: Find the constant of
proportionality.
Answer:
(b) Problem: Find x when y = 124.53.
Answer:
Key Points
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Linear Functions
Application: Straight-Line
Depreciation
Scatter Diagrams
Linear and Nonlinear Relationships
Line of Best Fit
Direct Variation
Library of Functions;
Piecewise-defined
Functions
Section 2.5
Linear Functions

f(x) = mx+b, m and
b a real number

Domain: ({1, 1)

Range: ({1, 1)
unless m = 0

Increasing on ({1, 1)
(if m > 0)

Decreasing on ({1, 1)
(if m < 0)

Constant on ({1, 1)
(if m = 0)
Constant Function

f(x) = b, b a real
number







Special linear functions
Domain: ({1, 1)
Range: fbg
Even/odd/neither: Even
(also odd if b = 0)
Constant on ({1, 1)
x-intercepts: None
(unless b = 0)
y-intercept: y = b.
Identity Function

f(x) = x







Special linear function
Domain: ({1, 1)
Range: ({1, 1)
Even/odd/neither: Odd
Increasing on ({1, 1)
x-intercepts: x = 0
y-intercept: y = 0.
Square Function

f(x) = x2







Domain: ({1, 1)
Range: [0, 1)
Even/odd/neither:
Even
Increasing on (0, 1)
Decreasing on ({1, 0)
x-intercepts: x = 0
y-intercept: y = 0.
Cube Function

f(x) = x3






Domain: ({1, 1)
Range: ({1, 1)
Even/odd/neither:
Odd
Increasing on ({1, 1)
x-intercepts: x = 0
y-intercept: y = 0.
Square Root Function







Domain: [0, 1)
Range: [0, 1)
Even/odd/neither:
Neither
Increasing on (0, 1)
x-intercepts: x = 0
y-intercept: y = 0
Cube Root Function







Domain: ({1, 1)
Range: ({1, 1)
Even/odd/neither: Odd
Increasing on ({1, 1)
x-intercepts: x = 0
y-intercept: y = 0
Reciprocal Function







Domain: x  0
Range: x  0
Even/odd/neither:
Odd
Decreasing on
({1, 0) [ (0, 1)
x-intercepts: None
y-intercept: None
Absolute Value Function

f(x) = jxj







Domain: ({1, 1)
Range: [0, 1)
Even/odd/neither:
Even
Increasing on (0, 1)
Decreasing on ({1, 0)
x-intercepts: x = 0
y-intercept: y = 0
Absolute Value Function

Can also write the absolute value
function as

This is a piecewise-defined function.
Greatest Integer Function

f(x) = int(x)






greatest integer less
than or equal to x
Domain: ({1, 1)
Range: Integers (Z)
Even/odd/neither:
Neither
y-intercept: y = 0
Called a step
function
Greatest Integer Function
Piecewise-defined Functions

Example. We can
define a function
differently on
different parts of its
domain.
(a) Problem: Find f({2)
Answer:
(b) Problem: Find f({1)
Answer:
(c) Problem: Find f(2)
Answer:
(d) Problem: Find f(3)
Answer:
8
6
4
2
-7.5
-5
-2.5
2.5
-2
-4
-6
-8
5
7.5
Key Points




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


Linear Functions
Constant Function
Identity Function
Square Function
Cube Function
Square Root Function
Cube Root Function
Reciprocal Function
Absolute Value Function
Key Points (cont.)


Greatest Integer Function
Piecewise-defined Functions
Graphing
Techniques:
Transformations
Section 2.6
Transformations


Use basic library of functions and
transformations to plot many other
functions.
Plot graphs that look “almost” like
one of the basic functions.
Shifts

Example.
Problem: Plot f(x) = x3, g(x) = x3 { 1
and h(x) = x3 + 2 on the same axes
Answer:
4
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Shifts

Vertical shift:
A real number k is added to the right
side of a function y = f(x),
 New function y = f(x) + k
 Graph of new function:

Graph of f shifted vertically up k units
(if k > 0)
 Down jkj units (if k < 0)

Shifts

Example.
4
Problem: Use the
graph of f(x) = jxj
to obtain the graph
of g(x) = jxj + 2
Answer:
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Shifts

Example.
Problem: Plot f(x) = x3, g(x) = (x { 1)3
and h(x) = (x + 2)3 on the same axes
Answer:
4
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Shifts

Horizontal shift:
Argument x of a function f is replaced
by x { h,
 New function y = f(x { h)
 Graph of new function:

Graph of f shifted horizontally right h
units (if h > 0)
 Left jhj units (if h < 0)
 Also y = f(x + h) in latter case

Shifts

Example.
4
Problem: Use the
graph of f(x) = jxj
to obtain the graph
of g(x) = jx+2j
Answer:
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Shifts

Example.
Problem: The graph of
a function y = f(x) is
given. Use it to plot
g(x) = f(x { 3) + 2
Answer:
4
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Compressions and Stretches

Example.
Problem: Plot f(x) = x3, g(x) = 2x3 and
on the same axes
Answer:
4
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Compressions and Stretches

Vertical compression/stretch:
Right side of function y = f(x) is
multiplied by a positive number a,
 New function y = af(x)
 Graph of new function:

Multiply each y-coordinate on the graph of
y = f(x) by a.
 Vertically compressed (if 0 < a < 1)
 Vertically stretched (if a > 1)

Compressions and Stretches

Example.
Problem: Use the
graph of f(x) = x2
to obtain the graph
of g(x) = 3x2
Answer:
4
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Compressions and Stretches

Example.
Problem: Plot f(x) = x3, g(x) = (2x)3
and
on the same axes
4
Answer:
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Compressions and Stretches

Horizontal compression/stretch:
Argument x of a function y = f(x) is
multiplied by a positive number a
 New function y = f(ax)
 Graph of new function:

Divide each x-coordinate on the graph of
y = f(x) by a.
 Horizontally compressed (if a > 1)
 Horizontally stretched (if 0 < a < 1)

Compressions and Stretches

Example.
Problem: Use the
graph of f(x) = x2
to obtain the graph
of g(x) = (3x)2
Answer:
4
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Compressions and Stretches

Example.
4
Problem: The graph of
a function y = f(x) is
given. Use it to plot
g(x) = 3f(2x)
Answer:
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Reflections

Example.
Problem: f(x) = x3 + 1 and
g(x) = {(x3 + 1) on the same axes
4
Answer:
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Reflections

Reflections about x-axis :
Right side of the function
y = f(x) is multiplied by {1,
 New function y = {f(x)
 Graph of new function:


Reflection about the x-axis of the graph of
the function y = f(x).
Reflections

Example.
Problem: f(x) = x3 + 1 and
g(x) = ({x)3 + 1 on the same axes
4
Answer:
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Reflections

Reflections about y-axis :
Argument of the function
y = f(x) is multiplied by {1,
 New function y = f({x)
 Graph of new function:


Reflection about the y-axis of the graph of
the function y = f(x).
Summary of
Transformations
Summary of
Transformations
Summary of
Transformations
Summary of Transformations

Example.
Problem: Use transformations to graph the
function
Answer:
4
3
2
1
-4
-2
2
-1
-2
-3
-4
4
Key Points





Transformations
Shifts
Compressions and Stretches
Reflections
Summary of Transformations
Mathematical
Models: Constructing
Functions
Section 2.7
Mathematical Models

Example.
Problem: The volume V of a right circular
cylinder is V = ¼r2h. If the height is
three times the radius, express the
volume V as a function of r.
Answer:
Mathematical Models

Example. Anne has 5000 feet of
fencing available to enclose a
rectangular field. One side of the field
lies along a river, so only three sides
require fencing.
(a) Problem: Express the area A of the
rectangle as a function of x, where x is
the length of the side parallel to the
river.
Answer:
Mathematical Models

Example (cont.)
(b) Problem: Graph
A = A(x) and find
what value of x
makes the area
largest.
Answer:
(c) Problem: What
value of x makes
the area largest?
Answer:
3.5 10
3 10
2.5 10
2 10
1.5 10
1 10
6
6
6
6
6
6
500000
1000
2000
3000
4000
5000
6000
Key Points

Mathematical Models