Lesson 2: A Catalog of Essential Functions

Section 2.2A Catalogue of Essential Functions

V63.0121.021/041, Calculus I

New York University

September 8, 2010

Announcements

I First WebAssign-ments are due September 13I First written assignment is due September 15I Do the Get-to-Know-You survey for extra credit!

. . . . . .

. . . . . .

Announcements

I First WebAssign-ments aredue September 13

I First written assignment isdue September 15

I Do the Get-to-Know-Yousurvey for extra credit!

V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 2 / 31

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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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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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Outline

Linear functions

Other Polynomial functions

Other power functions

Rational functions

Trigonometric Functions

Exponential and Logarithmic functions

Transformations of Functions

Compositions of 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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Outline

Linear functions



Rational functions






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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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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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Outline

Linear functions



Rational functions






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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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Outline

Linear functions



Rational functions






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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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Outline

Linear functions



Rational functions






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I Sine and cosineI Tangent and cotangentI Secant and cosecant


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Outline

Linear functions



Rational functions






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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

Linear functions



Rational functions






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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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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

Linear functions



Rational functions






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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 There are many classes of algebraic functionsI Algebraic rules can be used to sketch graphs


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Lesson 2: A Catalog of Essential Functions