Lesson 1: Functions

Section 1.1–1.2
Functions

V63.0121.006/016, Calculus I

New York University

May 17, 2010

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Get your WebAssign accounts and do the Intro assignment. Class
Key: nyu 0127 7953
Office Hours: TBD

. . . . . .

Announcements

Get your WebAssign
accounts and do the Intro
assignment. Class Key:
nyu 0127 7953
Office Hours: TBD

. . . . . .

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

Objectives: Functions and their Representations

Understand the definition
of function.
Work with functions
represented in different
ways
Work with functions
defined piecewise over
several intervals.
Understand and apply the
definition of increasing and
decreasing function.

. . . . . .


Objectives: A Catalog of Essential Functions
Identify different classes of
algebraic functions,
including polynomial
(linear, quadratic, cubic,
etc.), polynomial
(especially linear,
quadratic, and cubic),
rational, power,
trigonometric, and
exponential functions.
Understand the effect of
algebraic transformations
on the graph of a function.
Understand and compute
the composition of two
functions. . . . . . .


What is a function?

Definition
A function f is a relation 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.

. . . . . .


Outline
.
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
.
. . . . . .


The Modeling Process

. .
Real-world
.
. m
. odel Mathematical
.
Problems Model

s
. olve
.est
t

. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions

. . . . . .


Plato's Cave

. . . . . .


The Modeling Process

. .
Real-world
.
. m
. odel Mathematical
.
Problems Model

s
. olve
.est
t

. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions

S
. hadows F
. orms
. . . . . .


Outline
.
Modeling
Monotonicity
Symmetry
Linear functions
Rational functions
.
. . . . . .



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 after
substitution, give a real number.

. . . . . .


Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2

. . . . . .


Example
x+1
x−2

Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2.

. . . . . .


Example
x+1
x−2

Solution
except 2. As for the range, we can solve

x+2 2y + 1
y= =⇒ x =
x−1 y−1

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

. . . . . .


Example
x+1
x−2

Solution
except 2. As for the range, we can solve

x+2 2y + 1
y= =⇒ x =
x−1 y−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 }

. . . . . .


No-no's for expressions

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

. . . . . .


Piecewise-defined functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.

. . . . . .


Piecewise-defined functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.

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

. .
2 .

. .
1 . .

. . .
0
. 1
. 2
.
. . . . . .



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

. . . . . .


Example

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

x f(x)
1 4
2 5
3 6

. . . . . .


Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6

. . . . . .


Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6

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

. . . . . .


Example


x f(x)
1 4
2 4
3 6

. . . . . .


Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 4
3 6
. .
3 ..
6

. . . . . .


Example


. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 4
3 6
. .
3 ..
6

Yes, the range is {4, 6}.

. . . . . .


Example

How about this one?

x f(x)
1 4
1 5
3 6

. . . . . .


Example

How about this one?

. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
1 5
3 6
. .
3 ..
6

. . . . . .


Example

How about this one?

. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
1 5
3 6
. .
3 ..
6

No, that one’s not “deterministic.”

. . . . . .


An ideal function

. . . . . .

Why numerical functions matter

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

. . . . . .


Numerical Function Example

Here is the temperature in Boise, Idaho measured in 15-minute
intervals over the period August 22–29, 2008.
.
1
. 00 .
9
.0.
8
.0.
7
.0.
6
.0.
5
.0.
4
.0.
3
.0.
2
.0.
1
.0. . . . . . . .
8
. /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29
8 8 8 8 8 8 8

. . . . . .



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

.

.

. . . . . .



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

.

.

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



Oftentimes our functions come out of nature and have verbal
descriptions:
The temperature T(t) in this room at time t.
The elevation h(θ) of the point on the equator at longitude θ.
The utility u(x) I derive by consuming x burritos.

. . . . . .


Outline
.
Modeling
Monotonicity
Symmetry
Linear functions
Rational functions
.
. . . . . .


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?

. . . . . .


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

. .5 .
0

. . .
$
.0 $
. 52,115 $
. 100K
. . . . . .


Monotonicity

Definition

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

. . . . . .


Examples

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

. . . . . .


Examples

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

Example
Obviously, the temperature in Boise is neither increasing nor
decreasing.

. . . . . .


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 black
hole.

. . . . . .


Possible Intensity Graph

y
. = I(x)

.
x
.

. . . . . .


Possible Gravity Graph
y
. = F(x)

.
x
.

. . . . . .


Definitions

Definition

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

. . . . . .


Examples

Even: constants, even powers, cosine
Odd: odd powers, sine, tangent
Neither: exp, log

. . . . . .


Outline
.
Modeling
Monotonicity
Symmetry
Linear functions
Rational functions
.
. . . . . .



linear functions, defined by slope an intercept, point and point, or
point and slope.
quadratic functions, cubic functions, power functions, polynomials
rational functions
trigonometric functions
exponential/logarithmic functions

. . . . . .


Linear functions

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

f(x) = mx + b.

. . . . . .


Linear functions


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 function of distance x traveled.

. . . . . .


Linear functions


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 function of distance x traveled.

Answer
If x is in miles and f(x) in dollars,

f(x) = 2.5 + 2x

. . . . . .


Example
Biologists have noticed that the chirping rate of crickets of a certain
species is related to temperature, and the relationship 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 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.

. . . . . .


Solution

. . . . . .


Solution

The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )

. . . . . .


Solution

has equation
y − y0 = m(x − x0 )

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

. . . . . .


Solution

has equation
y − y0 = m(x − x0 )

80 − 70 10 1
173 − 113 60 6
So an equation for T and N is

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

. . . . . .


Solution

has equation
y − y0 = m(x − x0 )

80 − 70 10 1
173 − 113 60 6
So an equation for T and N is

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

37
If N = 150, then T = + 70 = 76 1 ◦ F
6
6
. . . . . .



Quadratic functions take the form

f(x) = ax2 + bx + c

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

. . . . . .



Quadratic functions take the form

f(x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0, downward if
a < 0.
Cubic functions take the form

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

. . . . . .


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

. . . . . .


Example

Solution
The general equation is y = ax2 + bx + c.

. . . . . .


Example

Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:

3 = a · 02 + b · 0 + c
−1 = a · 22 + b · 2 + c
0 = a · 32 + b · 3 + c

. . . . . .


Example

Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation 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

. . . . . .


Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives

−12 = 12a + 6b
−6 = 18a + 6b

. . . . . .



−12 = 12a + 6b
−6 = 18a + 6b

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

. . . . . .



−12 = 12a + 6b
−6 = 18a + 6b

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

−12 = 12 + 6b =⇒ b = −4

. . . . . .



−12 = 12a + 6b
−6 = 18a + 6b

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

−12 = 12 + 6b =⇒ b = −4

So our equation is
y = x2 − 4x + 3

. . . . . .



Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.

. . . . . .


Rational functions

Definition
A rational function is a quotient of polynomials.

Example
x3 (x + 3)
The function f(x) = is rational.
(x + 2)(x − 1)

. . . . . .



Sine and cosine
Tangent and cotangent
Secant and cosecant

. . . . . .



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

. . . . . .


Outline
.
Modeling
Monotonicity
Symmetry
Linear functions
Rational functions
.
. . . . . .



Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1

. . . . . .



Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
Observe that if the fiddling occurs within the function, a transformation
is applied on the x-axis. After the function, to the y-axis.

. . . . . .


Vertical and Horizontal Shifts

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

. . . . . .



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

. . . . . .



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

. . . . . .



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

. . . . . .



y = f(x − c), shift the graph of y = f(x) a distance c units to the right
y = f(x + c), shift the graph of y = f(x) a distance c units to the left

. . . . . .


Now try these

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

. . . . . .


Scaling and flipping

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

. . . . . .



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

. . . . . .



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

. . . . . .



If |c| < 1, the scaling is a compression

. . . . . .



If |c| < 1, the scaling is a compression
If c < 0, the scaling includes a flip

. . . . . .


Outline
.
Modeling
Monotonicity
Symmetry
Linear functions
Rational functions
.
. . . . . .


Composition is a compounding of functions 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.

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

. . . . . .


Decomposing

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

Solution
√
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

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

. . . . . .


Lesson 1: Functions

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