We introduce a number of different functions that can be used for modeling.

1. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
.
Sec on 2.2 Notes
A Catalogue of Essen al Func ons
V63.0121.001, Calculus I
Professor Ma hew Leingang
New York University
Announcements
First WebAssign-ments are due January 31
First wri en assignment is due February 2
Do the Get-to-Know-You survey for extra credit!
. .
Announcements Notes
First WebAssign-ments
are due January 31
First wri en assignment
is due February 2
Do the Get-to-Know-You
survey for extra credit!
. .
Objectives Notes
Iden fy different classes of algebraic
func ons, including polynomial (linear,
quadra c, cubic, etc.),
polynomialra onal, power,
trigonometric, and exponen al
func ons.
Understand the effect of algebraic
transforma ons on the graph of a
func on.
Understand and compute the
composi on of two func ons.
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. 1
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2. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
What is a function? Notes
Defini on
A func on f is a rela on which assigns to to every element x in a set
D a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { y | y = f(x) for some x } is called the range of f.
. .
Classes of Functions Notes
linear func ons, defined by slope an intercept, point and point,
or point and slope.
quadra c func ons, cubic func ons, power func ons,
polynomials
ra onal func ons
trigonometric func ons
exponen al/logarithmic func ons
. .
Outline Notes
Algebraic Func ons
Linear func ons
Other Polynomial func ons
Other power func ons
General Ra onal func ons
Transcendental Func ons
Trigonometric Func ons
Exponen al and Logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
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3. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
Linear functions Notes
Linear 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.
Answer
If x is in miles and f(x) in dollars,
f(x) = 2.5 + 2x
. .
Notes
Example
Biologists have no ced that the chirping rate of crickets of a certain
species is related to temperature, and the rela onship appears to be
very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F
and 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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Solution Notes
Solu on
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. 3
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4. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
Other Polynomial functions Notes
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.
Cubic func ons take the form
f(x) = ax3 + bx2 + cx + d
. .
Other power functions Notes
Whole number powers: f(x) = xn .
1
nega ve powers are reciprocals: x−3 = 3 .
√ x
frac onal powers are roots: x1/3 = 3 x.
. .
General Rational functions Notes
Defini on
A ra onal func on is a quo ent of polynomials.
Example
x3 (x + 3)
The func on f(x) = is ra onal.
(x + 2)(x − 1)
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. 4
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5. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
Outline Notes
Algebraic Func ons
Linear func ons
Other Polynomial func ons
Other power func ons
General Ra onal func ons
Transcendental Func ons
Trigonometric Func ons
Exponen al and Logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
. .
Trigonometric Functions Notes
Sine and cosine
Tangent and cotangent
Secant and cosecant
. .
Trigonometric functions graphed Notes
GeoGebra applets
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. 5
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6. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
Exponential and Logarithmic Notes
functions
exponen al func ons (for example f(x) = 2x )
logarithmic func ons are their inverses (for example
f(x) = log2 (x))
. .
Graphs of exp and log Notes
GeoGebra applets
. .
Outline Notes
Algebraic Func ons
Linear func ons
Other Polynomial func ons
Other power func ons
General Ra onal func ons
Transcendental Func ons
Trigonometric Func ons
Exponen al and Logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
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. 6
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7. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
Transformations of Functions Notes
Take the squaring func on and graph these transforma ons:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
Observe that if the fiddling occurs within the func on, a
transforma on is applied on the x-axis. A er the func on, to the
y-axis.
. .
Vertical and Horizontal Shifts Notes
Suppose c > 0. To obtain the graph of
y = f(x) + c, shi the graph of y = f(x) a distance c units . . .
y = f(x) − c, shi the graph of y = f(x) a distance c units . . .
y = f(x − c), shi the graph of y = f(x) a distance c units . . .
y = f(x + c), shi the graph of y = f(x) a distance c units . . .
. .
Now try these Notes
y = sin (2x)
y = 2 sin (x)
y = e−x
y = −ex
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8. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
Scaling and flipping Notes
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f ver cally by c
If |c| < 1, the scaling is a compression
If c < 0, the scaling includes a flip
. .
Outline Notes
Algebraic Func ons
Linear func ons
Other Polynomial func ons
Other power func ons
General Ra onal func ons
Transcendental Func ons
Trigonometric Func ons
Exponen al and Logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
. .
Composition is a compounding of Notes
functions in succession
g◦f
x f . g (g ◦ f)(x)
f(x)
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9. . V63.0121.001, Calculus I
. Sec on 2.2 : Essen al Func ons
. January 26, 2011
Professor Ma hew Leingang
Composing Notes
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solu on
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same.
. .
Decomposing Notes
Example
√
Express x2 − 4 as a composi on of two func ons. What is its
domain?
Solu on
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Summary Notes
There are many classes of algebraic func ons
Algebraic rules can be used to sketch graphs
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. 9
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