Lesson 1: Functions

Section 1.1–1.2Functions

V63.0121.006/016, Calculus I

New York University

May 17, 2010

Announcements

I Get your WebAssign accounts and do the Intro assignment. ClassKey: nyu 0127 7953

I Office Hours: TBD

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. . . . . .

Announcements

I Get your WebAssignaccounts and do the Introassignment. Class Key:nyu 0127 7953

I Office Hours: TBD

V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 2 / 54

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Objectives: Functions and their Representations

I Understand the definitionof function.

I Work with functionsrepresented in differentways

I Work with functionsdefined piecewise overseveral intervals.

I Understand and apply thedefinition of increasing anddecreasing function.


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Objectives: A Catalog of Essential Functions

I Identify different classes ofalgebraic functions,including polynomial(linear, quadratic, cubic,etc.), polynomial(especially linear,quadratic, and cubic),rational, power,trigonometric, andexponential functions.

I Understand the effect ofalgebraic transformationson the graph of a function.

I Understand and computethe composition of twofunctions.


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What is a function?

DefinitionA function f is a relation which assigns to to every element x in a set Da single element f(x) in a set E.

I The set D is called the domain of f.I The set E is called the target of f.I The set { y | y = f(x) for some x } is called the range of f.


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Outline.

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ModelingExamples of functions

Functions expressed by formulasFunctions described numericallyFunctions described graphicallyFunctions described verbally

Properties of functionsMonotonicitySymmetry

Classes of FunctionsLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functions

Transformations of FunctionsCompositions of Functions


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The Modeling Process

...Real-worldProblems

..Mathematical

Model

..MathematicalConclusions

..Real-worldPredictions

.model.solve

.interpret

.test


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Plato's Cave


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The Modeling Process

...Real-worldProblems

..Mathematical

Model

..MathematicalConclusions

..Real-worldPredictions

.model.solve

.interpret

.test

.Shadows .Forms


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Outline.

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Functions expressed by formulas

Any expression in a single variable x defines a function. In this case,the domain is understood to be the largest set of x which aftersubstitution, give a real number.


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Formula function example

Example

Let f(x) =x+ 1x− 2

. Find the domain and range of f.

SolutionThe denominator is zero when x = 2, so the domain is all real numbersexcept 2. As for the range, we can solve

y =x+ 2x− 1

=⇒ x =2y+ 1y− 1

So as long as y ̸= 1, there is an x associated to y. Therefore

domain(f) = { x | x ̸= 2 }range(f) = { y | y ̸= 1 }


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Example

Let f(x) =x+ 1x− 2


SolutionThe denominator is zero when x = 2, so the domain is all real numbersexcept 2.

As for the range, we can solve

y =x+ 2x− 1

=⇒ x =2y+ 1y− 1




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Example

Let f(x) =x+ 1x− 2



y =x+ 2x− 1

=⇒ x =2y+ 1y− 1

So as long as y ̸= 1, there is an x associated to y.

Therefore



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Example

Let f(x) =x+ 1x− 2



y =x+ 2x− 1

=⇒ x =2y+ 1y− 1




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No-no's for expressions

I Cannot have zero in the denominator of an expressionI Cannot have a negative number under an even root (e.g., square

root)I Cannot have the logarithm of a negative number


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Piecewise-defined functions

Example

Let

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

Find the domain and range of f and graph the function.

SolutionThe domain is [0,2]. The range is [0,2). The graph is piecewise.

...0

..1

..2

..1

..2

.

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Piecewise-defined functions

Example

Let

f(x) =

{x2 0 ≤ x ≤ 1;3− x 1 < x ≤ 2.

Find the domain and range of f and graph the function.

SolutionThe domain is [0,2]. The range is [0,2). The graph is piecewise.

...0

..1

..2

..1

..2

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.


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Functions described numerically

We can just describe a function by a table of values, or a diagram.


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Example

Is this a function? If so, what is the range?

x f(x)1 42 53 6

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..1

..2

..3

. .4

. .5

. .6

Yes, the range is {4,5,6}.


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Example


x f(x)1 42 53 6

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..1

..2

..3

. .4

. .5

. .6



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Example


x f(x)1 42 53 6

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..1

..2

..3

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. .5

. .6



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Example


x f(x)1 42 43 6

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. .

..1

..2

..3

. .4

. .5

. .6

Yes, the range is {4,6}.


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Example


x f(x)1 42 43 6

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..1

..2

..3

. .4

. .5

. .6



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Example


x f(x)1 42 43 6

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..1

..2

..3

. .4

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Example

How about this one?

x f(x)1 41 53 6

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..1

..2

..3

. .4

. .5

. .6

No, that one’s not “deterministic.”


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Example

How about this one?

x f(x)1 41 53 6

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..1

..2

..3

. .4

. .5

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Example

How about this one?

x f(x)1 41 53 6

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..1

..2

..3

. .4

. .5

. .6



An ideal function

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Why numerical functions matter

In science, functions are often defined by data. Or, we observe dataand assume that it’s close to some nice continuous function.


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Numerical Function Example

Here is the temperature in Boise, Idaho measured in 15-minuteintervals over the period August 22–29, 2008.

...8/22

..8/23

..8/24

..8/25

..8/26

..8/27

..8/28

..8/29

..10

..20

..30

..40

..50

..60

..70

..80

..90

..100


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Functions described graphically

Sometimes all we have is the “picture” of a function, by which wemean, its graph.

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The one on the right is a relation but not a function.


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Functions described graphically

Sometimes all we have is the “picture” of a function, by which wemean, its graph.

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The one on the right is a relation but not a function.


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Functions described verbally

Oftentimes our functions come out of nature and have verbaldescriptions:

I The temperature T(t) in this room at time t.I The elevation h(θ) of the point on the equator at longitude θ.I The utility u(x) I derive by consuming x burritos.


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Outline.

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Monotonicity

Example

Let P(x) be the probability that my income was at least $x last year.What might a graph of P(x) look like?

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..1

..0.5

..$0

..$52,115

..$100K


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Monotonicity

Example

Let P(x) be the probability that my income was at least $x last year.What might a graph of P(x) look like?

.

..1

..0.5

..$0

..$52,115

..$100K


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Monotonicity

Definition

I A function f is decreasing if f(x1) > f(x2) whenever x1 < x2 forany two points x1 and x2 in the domain of f.

I A function f is increasing if f(x1) < f(x2) whenever x1 < x2 for anytwo points x1 and x2 in the domain of f.


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Examples

Example

Going back to the burrito function, would you call it increasing?

Example

Obviously, the temperature in Boise is neither increasing nordecreasing.


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Examples

Example

Going back to the burrito function, would you call it increasing?

Example

Obviously, the temperature in Boise is neither increasing nordecreasing.


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Symmetry

Example

Let I(x) be the intensity of light x distance from a point.

Example

Let F(x) be the gravitational force at a point x distance from a blackhole.


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Possible Intensity Graph

..x

.y = I(x)


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Possible Gravity Graph

..x

.y = F(x)


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Definitions

Definition

I A function f is called even if f(−x) = f(x) for all x in the domain of f.I A function f is called odd if f(−x) = −f(x) for all x in the domain of

f.


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Examples

I Even: constants, even powers, cosineI Odd: odd powers, sine, tangentI Neither: exp, log


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Outline.

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Classes of Functions

I linear functions, defined by slope an intercept, point and point, orpoint and slope.

I quadratic functions, cubic functions, power functions, polynomialsI rational functionsI trigonometric functionsI exponential/logarithmic functions


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Linear functions

Linear functions have a constant rate of growth and are of the form

f(x) = mx+ b.

Example

In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Writethe fare f(x) as a function of distance x traveled.

AnswerIf x is in miles and f(x) in dollars,

f(x) = 2.5+ 2x


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Linear functions


f(x) = mx+ b.

Example



f(x) = 2.5+ 2x


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Linear functions


f(x) = mx+ b.

Example



f(x) = 2.5+ 2x


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Example

Biologists have noticed that the chirping rate of crickets of a certainspecies is related to temperature, and the relationship appears to bevery nearly linear. A cricket produces 113 chirps per minute at 70 ◦Fand 173 chirps per minute at 80 ◦F.(a) Write a linear equation that models the temperature T as a function

of the number of chirps per minute N.(b) What is the slope of the graph? What does it represent?(c) If the crickets are chirping at 150 chirps per minute, estimate the

temperature.


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Solution

I The point-slope form of the equation for a line is appropriatehere: If a line passes through (x0, y0) with slope m, then the linehas equation

y− y0 = m(x− x0)

I The slope of our line is80− 70

173− 113=

1060

=16

I So an equation for T and N is

T− 70 =16(N− 113) =⇒ T =

16N− 113

6+ 70

I If N = 150, then T =376

+ 70 = 7616◦F


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Solution


y− y0 = m(x− x0)


173− 113=

1060

=16


T− 70 =16(N− 113) =⇒ T =

16N− 113

6+ 70

I If N = 150, then T =376

+ 70 = 7616◦F


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Solution


y− y0 = m(x− x0)


173− 113=

1060

=16


T− 70 =16(N− 113) =⇒ T =

16N− 113

6+ 70

I If N = 150, then T =376

+ 70 = 7616◦F


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Solution


y− y0 = m(x− x0)


173− 113=

1060

=16


T− 70 =16(N− 113) =⇒ T =

16N− 113

6+ 70

I If N = 150, then T =376

+ 70 = 7616◦F


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Solution


y− y0 = m(x− x0)


173− 113=

1060

=16


T− 70 =16(N− 113) =⇒ T =

16N− 113

6+ 70

I If N = 150, then T =376

+ 70 = 7616◦F


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Other Polynomial functions

I Quadratic functions take the form

f(x) = ax2 + bx+ c

The graph is a parabola which opens upward if a > 0, downward ifa < 0.

I Cubic functions take the form

f(x) = ax3 + bx2 + cx+ d


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Other Polynomial functions

I Quadratic functions take the form

f(x) = ax2 + bx+ c

The graph is a parabola which opens upward if a > 0, downward ifa < 0.

I Cubic functions take the form

f(x) = ax3 + bx2 + cx+ d


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Example

A parabola passes through (0,3), (3,0), and (2,−1). What is theequation of the parabola?

SolutionThe general equation is y = ax2 + bx+ c. Each point gives anequation relating a, b, and c:

3 = a · 02 + b · 0+ c

−1 = a · 22 + b · 2+ c

0 = a · 32 + b · 3+ c

Right away we see c = 3. The other two equations become

−4 = 4a+ 2b−3 = 9a+ 3b


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Example


SolutionThe general equation is y = ax2 + bx+ c.

Each point gives anequation relating a, b, and c:

3 = a · 02 + b · 0+ c

−1 = a · 22 + b · 2+ c

0 = a · 32 + b · 3+ c


−4 = 4a+ 2b−3 = 9a+ 3b


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Example



3 = a · 02 + b · 0+ c

−1 = a · 22 + b · 2+ c

0 = a · 32 + b · 3+ c


−4 = 4a+ 2b−3 = 9a+ 3b


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Example



3 = a · 02 + b · 0+ c

−1 = a · 22 + b · 2+ c

0 = a · 32 + b · 3+ c


−4 = 4a+ 2b−3 = 9a+ 3b


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Solution (Continued)

Multiplying the first equation by 3 and the second by 2 gives

−12 = 12a+ 6b−6 = 18a+ 6b

Subtract these two and we have −6 = −6a =⇒ a = 1. Substitutea = 1 into the first equation and we have

−12 = 12+ 6b =⇒ b = −4

So our equation isy = x2 − 4x+ 3


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−12 = 12a+ 6b−6 = 18a+ 6b

Subtract these two and we have −6 = −6a =⇒ a = 1.

Substitutea = 1 into the first equation and we have

−12 = 12+ 6b =⇒ b = −4



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−12 = 12a+ 6b−6 = 18a+ 6b


−12 = 12+ 6b =⇒ b = −4



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−12 = 12a+ 6b−6 = 18a+ 6b


−12 = 12+ 6b =⇒ b = −4



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Other power functions

I Whole number powers: f(x) = xn.

I negative powers are reciprocals: x−3 =1x3

.

I fractional powers are roots: x1/3 = 3√x.


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Rational functions

DefinitionA rational function is a quotient of polynomials.

Example

The function f(x) =x3(x+ 3)

(x+ 2)(x− 1)is rational.


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Trigonometric Functions

I Sine and cosineI Tangent and cotangentI Secant and cosecant


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Exponential and Logarithmic functions

I exponential functions (for example f(x) = 2x)I logarithmic functions are their inverses (for example f(x) = log2(x))


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Outline.

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Transformations of Functions

Take the squaring function and graph these transformations:

I y = (x+ 1)2

I y = (x− 1)2

I y = x2 + 1I y = x2 − 1

Observe that if the fiddling occurs within the function, a transformationis applied on the x-axis. After the function, to the y-axis.


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Transformations of Functions

Take the squaring function and graph these transformations:

I y = (x+ 1)2

I y = (x− 1)2

I y = x2 + 1I y = x2 − 1

Observe that if the fiddling occurs within the function, a transformationis applied on the x-axis. After the function, to the y-axis.


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Vertical and Horizontal Shifts

Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units

upward

I y = f(x)− c, shift the graph of y = f(x) a distance c units

downward

I y = f(x−c), shift the graph of y = f(x) a distance c units

to the right

I y = f(x+ c), shift the graph of y = f(x) a distance c units

to the left


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Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units

downward

I y = f(x−c), shift the graph of y = f(x) a distance c units

to the right


to the left


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Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units downwardI y = f(x−c), shift the graph of y = f(x) a distance c units

to the right


to the left


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Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units downwardI y = f(x−c), shift the graph of y = f(x) a distance c units to the rightI y = f(x+ c), shift the graph of y = f(x) a distance c units

to the left


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Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units downwardI y = f(x−c), shift the graph of y = f(x) a distance c units to the rightI y = f(x+ c), shift the graph of y = f(x) a distance c units to the left


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Now try these

I y = sin (2x)I y = 2 sin (x)I y = e−x

I y = −ex


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Scaling and flipping

To obtain the graph ofI y = f(c · x), scale the graph of f

horizontally

by cI y = c · f(x), scale the graph of f

vertically

by cI If |c| < 1, the scaling is a

compression

I If c < 0, the scaling includes a

flip


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To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f

vertically

by cI If |c| < 1, the scaling is a

compression


flip


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To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f vertically by cI If |c| < 1, the scaling is a

compression


flip


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To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f vertically by cI If |c| < 1, the scaling is a compressionI If c < 0, the scaling includes a

flip


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To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f vertically by cI If |c| < 1, the scaling is a compressionI If c < 0, the scaling includes a flip


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Outline.

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Composition is a compounding of functions in

succession

..f .g

.g ◦ f

.x .(g ◦ f)(x).f(x)

.


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Composing

Example

Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

Solutionf ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2). Note they are not the same.


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Composing

Example

Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

Solutionf ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2). Note they are not the same.


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Decomposing

Example

Express√x2 − 4 as a composition of two functions. What is its

domain?

SolutionWe can write the expression as f ◦ g, where f(u) =

√u and

g(x) = x2 − 4. The range of g needs to be within the domain of f. Toinsure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.


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Summary

I The fundamental unit of investigation in calculus is the function.I Functions can have many representationsI There are many classes of algebraic functionsI Algebraic rules can be used to sketch graphs


Education

Lesson 1: Functions