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Sec on 2.2A Catalogue of Essen al Func ons

V63.0121.001, Calculus IProfessor Ma hew Leingang

New York University

Announcements

I First WebAssign-ments are due January 31I First wri en assignment is due February 2I Do the Get-to-Know-You survey for extra credit!

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Announcements

I First WebAssign-mentsare due January 31

I First wri en assignmentis due February 2

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

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ObjectivesI Iden fy different classes of algebraicfunc ons, including polynomial (linear,quadra c, cubic, etc.),polynomialra onal, power,trigonometric, and exponen alfunc ons.

I Understand the effect of algebraictransforma ons on the graph of afunc on.

I Understand and compute thecomposi on of two func ons.

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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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What is a function?

Defini onA func on f is a rela on which assigns to to every element x in a setD a 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 func ons, defined by slope an intercept, point and point,or point and slope.

I quadra c func ons, cubic func ons, power func ons,polynomials

I ra onal func onsI trigonometric func onsI exponen al/logarithmic func ons

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OutlineAlgebraic Func ons

Linear func onsOther Polynomial func onsOther power func onsGeneral Ra onal func ons

Transcendental Func onsTrigonometric Func onsExponen al and Logarithmic func ons

Transforma ons of Func ons

Composi ons of Func ons

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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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Linear functionsLinear func ons 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.Write the fare f(x) as a func on of distance x traveled.

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

f(x) = 2.5+ 2x

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Example

Biologists have no ced that the chirping rate of crickets of a certainspecies is related to temperature, and the rela onship 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 equa on that models the temperature T as a

func on of the number of chirps per minute N.(b) If the crickets are chirping at 150 chirps per minute, es mate the

temperature.

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SolutionSolu on

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

y− y0 = m(x− x0)

I The slope of our line is80− 70

173− 113=

1060

=16

I So an equa on 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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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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Other Polynomial functionsI Quadra c func ons take the form

f(x) = ax2 + bx+ c

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

I Cubic func ons take the form

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

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Other power functions

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

I nega ve powers are reciprocals: x−3 =1x3.

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

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General Rational functions

Defini onA ra onal func on is a quo ent of polynomials.

Example

The func on f(x) =x3(x+ 3)

(x+ 2)(x− 1)is ra onal.

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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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OutlineAlgebraic Func ons

Linear func onsOther Polynomial func onsOther power func onsGeneral Ra onal func ons

Transcendental Func onsTrigonometric Func onsExponen al and Logarithmic func ons

Transforma ons of Func ons

Composi ons of Func ons

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Trigonometric Functions

I Sine and cosineI Tangent and cotangentI Secant and cosecant

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Trigonometric functions graphed

GeoGebra applets

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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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Exponential and Logarithmicfunctions

I exponen al func ons (for example f(x) = 2x)I logarithmic func ons are their inverses (for example

f(x) = log2(x))

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Graphs of exp and log

GeoGebra applets

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OutlineAlgebraic Func ons

Linear func onsOther Polynomial func onsOther power func onsGeneral Ra onal func ons

Transcendental Func onsTrigonometric Func onsExponen al and Logarithmic func ons

Transforma ons of Func ons

Composi ons of Func ons

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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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Transformations of FunctionsTake the squaring func on and graph these transforma ons:

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 func on, atransforma on is applied on the x-axis. A er the func on, to they-axis.

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Vertical and Horizontal ShiftsSuppose c > 0. To obtain the graph of

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

upward

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

downward

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

tothe right

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

tothe le

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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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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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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 ver cally by cI If |c| < 1, the scaling is a compressionI If c < 0, the scaling includes a flip

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OutlineAlgebraic Func ons

Linear func onsOther Polynomial func onsOther power func onsGeneral Ra onal func ons

Transcendental Func onsTrigonometric Func onsExponen al and Logarithmic func ons

Transforma ons of Func ons

Composi ons of Func ons

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Composition is a compounding offunctions in succession

..f . g.

g ◦ f

.x . (g ◦ f)(x).f(x).

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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011

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Composing

Example

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

Solu onf ◦ 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 composi on of two func ons. What is itsdomain?

Solu on

We 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 func onsI Algebraic rules can be used to sketch graphs

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. Sec on 2.2 : Essen al Func ons. V63.0121.001, Calculus IProfessor Ma hew Leingang

. January 26, 2011