Lesson 2: A Catalog of Essential Functions (slides)

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Sec on 2.2
A Catalogue of Essen al Func ons
V63.0121.011, 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
First recita ons are February 3

Announcements

First WebAssign-ments
are due January 31
First wri en assignment
is due February 2
First recita ons are
February 3

Objectives
Iden fy diﬀerent classes of algebraic
func ons, including polynomial
(linear,quadra c,cubic, etc.), ra 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.

Recall: What is a function?

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.

Four ways to represent a function

verbally—by a descrip on in words
numerically—by a table of values or a list of data
visually—by a graph
symbolically or algebraically—by a formula
Today the focus is on the diﬀerent kinds of formulas that can be
used to represent func ons.

Classes of Functions

linear func ons, deﬁned by slope and intercept, two points, 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
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

Linear functions
Linear func ons have a y
constant rate of growth and (x2 , y2 )
are of the form
(x1 , y1 ) ∆y = y2 − y1
f(x) = mx + b. (0, b)
∆x = x2 − x1
The slope m represents the ∆y
m=
“steepness” of the graphed ∆x
line, and the intercept b
. x
represents an ini al value of
the func on.

Modeling with Linear Functions
Example
Assume that a taxi costs $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.

Modeling with Linear Functions
Example
Assume that a taxi costs $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
The ini al fare is $2.50, and the change in fare per mile is
$0.40/0.2 mi = $2/mi. So if x is in miles and f(x) in dollars, the
equa on is
f(x) = 2.5 + 2x

A Biological Example
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.

Biological Example: Solution
Solu on
The point-slope form of the equa on for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equa on y − y0 = m(x − x0 ).

Solu on
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6

Solu on
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equa on rela ng T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6

Solution continued
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6

Solution continued
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
If N = 150, then
37
T= + 70 = 76 1 ◦ F
6
6

Other polynomial functions
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.

Other polynomial functions
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

Whole number powers: f(x) = xn .
1
nega ve powers are reciprocals: x−3 = 3 .
1/3
√ x
frac onal powers are roots: x = 3 x.

General rational functions
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)
The domain is all real numbers except −2 and 1.
The func on is 0 when x = 0 or x = −3.

Trigonometric functions

Sine and cosine
Tangent and cotangent
Secant and cosecant
GeoGebra applets to graph these

Exponential and logarithmic
functions

exponen al func ons (for example f(x) = 2x )
logarithmic func ons are their inverses (for example
f(x) = log2 (x))
GeoGebra applets to graph these

Transformations of Functions
Take the squaring func on and graph these transforma ons:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1

Transformations of Functions
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
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 . . .

upward



upward
downward


upward
downward
y = f(x − c), shi the graph of y = f(x) a distance c units . . . to
the right

upward
downward
y = f(x − c), shi the graph of y = f(x) a distance c units . . . to
the right
y = f(x + c), shi the graph of y = f(x) a distance c units . . . to
the le

Why?
Ques on
Why is the graph of g(x) = f(x + c) a shi of the graph of f(x) to the
le by c?

Why?
Ques on
Why is the graph of g(x) = f(x + c) a shi of the graph of f(x) to the
le by c?

Answer
Think about x as me. Then x + c is the me c into the future. To
rec fy the future of the graph of f with that of g, pull the graph of f c
into the past.

Illustrating the shift

(x, f(x))

.
x


Adding c moves x to the
right (x, f(x))

.
x x+c

(x + c, f(x + c))

right (x, f(x))
But then f is applied

.
x x+c

(x, f(x + c)) (x + c, f(x + c))

right (x, f(x))
To get the graph of
f(x + c), the value
f(x + c) must be above x
.
x x+c

(x, f(x + c)) (x + c, f(x + c))

right (x, f(x))
To get the graph of
f(x + c), the value
f(x + c) must be above x
So we translate backward .
x x+c

Now try these

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

Scaling and flipping
c<0 c>0
|c| > 1 |c| < 1 |c| < 1 |c| > 1

f(cx) . . . .

H compress, flip H stretch, flip H stretch H compress

cf(x) . . . .

V stretch, flip V compress, flip V compress V stretch

Composition of Functions
Compounding in Succession

g◦f
x f . g (g ◦ f)(x)
f(x)

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

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

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

Decomposing
Example
√
Express x2 − 4 as a composi on of two func ons. What is its
domain?

Decomposing
Example
√
Express x2 − 4 as a composi on of two func ons. What is its
domain?

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. To
insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.

Summary

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

Lesson 2: A Catalog of Essential Functions (slides)

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