functions and their graphs

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.

Melissa

John

Jennifer

Patrick

$45,000

$40,000

$50,000

Person

Salary

Relations

Example.

0

1

4

0

1

{1

2

{2

Number

Number

Functions

Function: special kind of relationEach 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:

Melissa

John

Jennifer

Patrick

$45,000

$40,000

$50,000

Person

Salary

Functions

Example.Problem: Does this relation represent

a function?Answer:

0

1

4

0

1

{1

2

{2

Number

Number

Domain and Range

Function from X to YDomain 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:

0

1

4

9

{3

{2

{1

0

1

2

3

X Y

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 functionSolve 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 numberWe 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


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

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

)()(

)(xgxf

xgf


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:


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

RelationsFunctionsDomain and RangeEquations as FunctionsFunction as a MachineFinding Values of a Function Implicit Form of a Function Important FactsFinding the Domain

Key Points (cont.)


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

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Example.Problem: Is the graph that of a

function?Answer:

Finding Information From Graphs

Example. Answer the questions about the graph.

(a) Problem: Find f(0)

Answer:

(b) Problem: Find f(2)

Answer:

(c) Problem: Find the

domain

Answer:

(d) Problem: Find the range

Answer:

-4 -2 2 4

-4

-2

2

4

2, 452, 4

5 1,21,20,4


Example. Answer the questions about the graph.

(e) Problem: Find the x-intercepts:

Answer:

(f) Problem: Find the y-intercepts:

Answer:

-4 -2 2 4

-4

-2

2

4

2, 452, 4

5 1,21,20,4


Example. Answer the questions about the graph.(g) Problem: How often

does the line y = 3 intersect the graph?

Answer:

(h) Problem: For what values of x does f(x) = 2?

Answer:

(i) Problem: For what values of x is f(x) > 0?

Answer:

-4 -2 2 4

-4

-2

2

4

2, 452, 4

5 1,21,20,4

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

Vertical-line TestFinding Information From

GraphsFinding Information From

Formulas

Properties of Functions

Section 2.3

Even and Odd Functions

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)


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.


Example. Problem:

Does the graph represent a function which is even, odd, or neither?

Answer:

-4 -2 2 4

-4

-2

2

4

Identifying Even and Odd Functions from

the EquationExample. 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): For any choice of x1 and x2 in I, with

x1 < x2, we have f(x1) < f(x2) 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) 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 2 4 6

-6

-4

-2

2

4

6

Increasing, Decreasing and Constant FunctionsWARNING!

Describe the behavior of a graph in terms of its x-values.

Answers for these questions should be open intervals.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Local Extrema Local maximum 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. 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 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.

(a) Problem: At which

number(s) does f

have a local

maximum?

Answer:

(b) Problem: At which

number(s) does f

have a local

minimum?

Answer:

-7.5 -5 -2.5 2.5 5 7.5

-6

-4

-2

2

4

6

Average Rate of Change

Slope of a line can be interpreted as the average rate of changeAverage 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 LinesGeometric interpretation to the

average rate of changeLabel 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

-7.5 -5 -2.5 2.5 5 7.5

-5

-2.5

2.5

5

7.5

10

12.5

15

Secant Lines Example.

Problem: Find an

equation of the

secant line to

containing (0,

f(0)) and (5,

f(5))

Answer:

Key Points

Even and Odd FunctionsDescription of Even and Odd

Functions Identifying Even and Odd

Functions from the Equation Increasing, Decreasing and

Constant FunctionsLocal ExtremaAverage 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

-10 -5 5 10

-10

-7.5

-5

-2.5

2.5

5

7.5

10

Linear Functions

Example. Problem: Graph

the linear functionf(x) = 2x { 5

Answer:

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.

2 4 6 8 10 12 14

-40000

-20000

20000

40000

60000

80000

100000

120000

140000

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:


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

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

6 .0

7 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

0 .0

1 .0

2 .0

3 .0

4 .0

5 .0

6 .0

7 .0

8 .0

9 .0

1 0 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 0 .0

-8 .0

-6 .0

-4 .0

-2 .0

0 .0

2 .0

4 .0

6 .0

8 .0

1 0 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 2 .0

-1 0 .0

-8 .0

-6 .0

-4 .0

-2 .0

0 .0

2 .0

4 .0

6 .0

8 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 5 .0

-1 0 .0

-5 .0

0 .0

5 .0

1 0 .0

1 5 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

-1 5 .0

-1 0 .0

-5 .0

0 .0

5 .0

1 0 .0

1 5 .0

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5

Linear Nonlinear

Linear

Nonlinear Linear Nonlinear

Line of Best Fit

For linearly related scatter diagramLine 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

Linear FunctionsApplication: Straight-Line

DepreciationScatter DiagramsLinear and Nonlinear

RelationshipsLine of Best FitDirect 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


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


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

-7.5 -5 -2.5 2.5 5 7.5

-8

-6

-4

-2

2

4

6

8

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:

Key Points

Linear FunctionsConstant Function Identity FunctionSquare FunctionCube FunctionSquare Root FunctionCube Root FunctionReciprocal FunctionAbsolute Value Function

Key Points (cont.)

Greatest Integer FunctionPiecewise-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 -2 2 4

-4

-3

-2

-1

1

2

3

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)

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts

Example. Problem: Use

the graph of f(x) = jxj to obtain the graph of g(x) = jxj + 2

Answer:

Shifts

Example. Problem: Plot f(x) = x3, g(x) = (x

{ 1)3 and h(x) = (x + 2)3 on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

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

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts


the graph of f(x) = jxj to obtain the graph of g(x) = jx+2j

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

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:

Compressions and Stretches

Example. Problem: Plot f(x) = x3, g(x) =

2x3 and on the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


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)



the graph of f(x) = x2 to obtain the graph of g(x) = 3x2

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Example.

Problem: Plot f(x) = x3, g(x) =

(2x)3

and on the same

axes

Answer:-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


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)

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4



the graph of f(x) = x2 to obtain the graph of g(x) = (3x)2

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4


Example. Problem: The

graph of a function y = f(x) is given. Use it to plot g(x) = 3f(2x)

Answer:

Reflections

Example.

Problem: f(x) = x3 + 1 and

g(x) = {(x3 + 1) on the same

axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

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

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

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

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Summary of Transformations

Example. Problem: Use transformations to

graph the functionAnswer:

Key Points

TransformationsShiftsCompressions and StretchesReflectionsSummary 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:

1000 2000 3000 4000 5000 6000

500000

1106

1.5106

2106

2.5106

3106

3.5106

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:

Key Points

Mathematical Models

functions and their graphs

Documents

function of x

function problem

element of x

function fx

image of x

terms of x

value of x

xy x