Lecture 1.2 Functions

8/7/2019 Lecture 1.2 Functions

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Functionsand Their

GraphsChapter 1.2



Functions

Section 1.2.1



Relations

Relation: A correspondencebetween 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

Perso

n

Salary



Relations

Example.

0

1

4

0

1

{1

2

{2

Numbe

r

Numbe

r



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 relationthat 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 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 ycorresponding to x

Range of the function: the set of allvalues 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 therelation represents a function. If it

is a function, state the domain andrange.

Problem:

Relation: f(2,5), (6,3), (8,2), (4,3)g

Answer:



Domain and Range

Example. Determine whether therelation represents a function. If it

is a function, state the domain andrange.

Problem:

Relation: f(1,7), (0, {3), (2,4), (1,8)g

Answer:



Equations as Functions

To determine whether an equationis a function

Solve the equation for y. If any value of x in the domain

corresponds to more than one y, theequation doesn¶t define a function

Otherwise, it does define a function.



Equations as Functions

Example.

Problem: Determine if the equation

x + y2

= 9defines y as a function of x.

Answer:



Function as a Machine

Accepts numbers from domain asinput.

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:



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 isgiven 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 thedomain

We usually call x the independentvariable

y is the dependent variable



Finding the Domain

If the domain isn¶t specified, it willalways be the largest set of real

numbers for which f(x) is a realnumber

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:



Finding the Domain

Example. A rectangular gardenhas 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:



The Graph of aFunction

Section 1.2.2



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:



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 thegraph.

(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

H2, 4¼¼¼¼¼5H- 2, 4¼¼¼¼¼5L H1,2

LH- 1,2

L

0,4






(e) Problem: Find thex-intercepts:

Answer:

(f) Problem: Find the

y-intercepts:Answer:

-4 -2 2 4

-4

-2

2

4

H2, 4¼¼¼¼¼5H- 2, 4¼¼¼¼¼5L H1,2

LH- 1,2

L

0,4






(g) Problem: How oftendoes 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

H2, 4¼¼¼¼¼5H- 2, 4¼¼¼¼¼5L H1,2

LH- 1,2

L

0,4




Formulas Example. Answer the following

questions for the functionf(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 isthe corresponding point on the graph?

Answer:(c) Problem: If f(x) = 1, what is x? What is

(are) the corresponding point(s) on thegraph?

Answer:



Key Points

Vertical-line Test

Finding Information From Graphs

Finding Information FromFormulas



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)



Description of Even and

Odd Functions Theorem.

A function is even if and only if its

graph is symmetric with respect tothe y-axis.A function is odd if and only if itsgraph is symmetric with respect to

the origin.



Description of Even and Odd

Functions Example.

Problem: Doesthe graphrepresent afunctionwhich is

even, odd, or neither?

Answer:

-4 -2 2 4

-4

-2

2

4




Functions Example.



Answer:

-4 -2 2 4

-4

-2

2

4




Functions Example.



Answer:

-4 -2 2 4

-4

-2

2

4



Identifying Even and OddFunctions 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): 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 intervalI) 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 thefunction shown.

(a) Problem: Where is thefunction increasing?

Answer:

(b) Problem: Where is thefunction 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 thebehavior of a graph

in terms of its x-values.

Answers for thesequestions shouldbe 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 allx · 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 allx · 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.



Average Rate of Change

Slope of a line can be interpretedas the average rate of change

Average rate of change: If c is in thedomain 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 theaverage 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 afunction equals the slope of thesecant 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:



Linear

Functions and

ModelsSection 1.2.4



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 constantaverage 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 thelinear function

f(x) = 2x { 5Answer:



Application: Straight-Line

Depreciation Example. Suppose that a company

has just purchased a new machinefor its manufacturing facility for $120,000. The company choosesto depreciate the machine usingthe straight-line method over 10

years.For straight-line depreciation, thevalue 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 functionthat expresses the book value of themachine as a function of its age, x

Answer:

(b) Problem: Graph the linear function

Answer:


Depreciation



Example. (cont.)

(c) Problem: What is the book value of the machine after 4 years?

Answer:

(d) Problem: When will the machine beworth $20,000?

Answer:


Depreciation



Library of

Functions;Piecewise-defined

FunctionsSection 1.2.5



Linear Functions

f(x) = mx+b, m and ba 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 realnumber

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


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)



x-intercepts: x = 0

y-intercept: y = 0.



Square Root Function

Domain: [0, 1)

Range: [0, 1)

Even/odd/neither:Neither


x-intercepts: x = 0

y-intercept: y = 0



Cube Root Function

Domain: ({1, 1)

Range: ({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


Decreasing on ({1, 0)

x-intercepts: x = 0

y-intercept: y = 0



Absolute Value Function

Can also write the absolute valuefunction 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 -

.5

.5 5 7.5

-8

-6

-4

-

4

6

Piecewise-defined

Functions Example. We candefine a functiondifferently ondifferent 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:

raphing



raphing

Techniques:Transformation

sSection 2.6



Transformations

Use basic library of functions andtransformations to plot manyother functions.

Plot graphs that look ³almost´ likeone of the basic functions.



Shifts

Example.

Problem: Plot f(x) = x3, g(x) = x3 { 1 andh(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 rightside 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 thegraph of f(x) = jxj

to obtain thegraph 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 isreplaced by x { h,

New function y = f(x { h)


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

Example.

Problem: Use thegraph of f(x) = jxj

to obtain thegraph of g(x) = jx+2j

Answer:



-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Shifts

Example.

Problem: The graphof a function y =

f(x) is given. Use itto plotg(x) = f(x { 3) + 2

Answer:

C i d



Compressions and

Stretches Example.

Problem: Plot f(x) = x3, g(x) = 2x3 andon the same axes

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

C i d



Compressions and

Stretches Vertical compression/stretch:

Right side of function y = f(x) ismultiplied by a positive number a,

New function y = af(x)


Multiply each y-coordinate on the graph

of y = f(x) by a.

Vertically compressed (if 0 < a < 1)

Vertically stretched (if a > 1)

C i d



Compressions and

Stretches Example.

Problem: Use thegraph of f(x) = x2

to obtain thegraph of g(x) =3x2

Answer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

C i d



Compressions and

Stretches Example.

Problem: Plot f(x) = x3, g(x) = (2x)3

and on the same axesAnswer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

C i d



Compressions and

Stretches Horizontal compression/stretch:

Argument x of a function y = f(x) ismultiplied by a positive number a

New function y = f(ax)


Divide each x-coordinate on the graph of

y = f(x) by a. Horizontally compressed (if a > 1)

Horizontally stretched (if 0 < a < 1)

C i d



-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Compressions and

Stretches Example.

Problem: Use thegraph of f(x) = x2

to obtain thegraph of g(x) =(3x)2

Answer:

C i d



-4 -2 2 4

-4

-3

-2

-1

1

2

3

4

Compressions and

Stretches Example.

Problem: The graphof a function y =

f(x) is given. Use itto plotg(x) = 3f(2x)

Answer:



Reflections

Example.

Problem: f(x) = x3 + 1 and

g(x) = {(x3

+ 1) on the same axesAnswer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4



Reflections

Reflections about x-axis :

Right side of the functiony = f(x) is multiplied by {1,

New function y = {f(x)


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 axesAnswer:

-4 -2 2 4

-4

-3

-2

-1

1

2

3

4



Reflections

Reflections about y-axis :

Argument of the functiony = f(x) is multiplied by {1,

New function y = f({x)


Reflection about the y-axis of the graph

of the function y = f(x).

Summary of



Summary of

Transformations

Summary of



Summary of

Transformations



Mathematical



Mathematical

Models:Constructing

FunctionsSection 1.2.7





Mathematical Models

Example. Anne has 5000 feet of fencing available to enclose arectangular field. One side of the

field lies along a river, so onlythree 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 tothe river.

Answer:



1000 2000 3000 4000 5000 6000

500000

1́ 106

1.5´106

2́ 106

2.5´106

3́ 106

3.5´106

Mathematical Models

Example (cont.)

(b) Problem: GraphA = A(x) and find

what value of xmakes the arealargest.

Answer:

(c) Problem: Whatvalue of x makesthe area largest?

Answer:

Lecture 1.2 Functions

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