Section 1.1Functions and their Representations
V63.0121.021/041, Calculus I
New York University
September 8, 2010
Announcements
I First WebAssign-ments are due September 13
I First written assignment is due September 15
I 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 1.1 Functions September 8, 2010 2 / 33
Function
Objectives: Functions and their Representations
I Understand the definition offunction.
I Work with functionsrepresented in different ways
I Work with functions definedpiecewise over severalintervals.
I Understand and apply thedefinition of increasing anddecreasing function.
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What is a function?
Definition
A function f is a relation which assigns to to every element x in a set D asingle 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 .
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 4 / 33
Outline
Modeling
Examples of functionsFunctions expressed by formulasFunctions described numericallyFunctions described graphicallyFunctions described verbally
Properties of functionsMonotonicitySymmetry
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The Modeling Process
Real-worldProblems
MathematicalModel
MathematicalConclusions
Real-worldPredictions
modelsolve
interpret
test
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Plato’s Cave
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 7 / 33
The Modeling Process
Real-worldProblems
MathematicalModel
MathematicalConclusions
Real-worldPredictions
modelsolve
interpret
test
Shadows Forms
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 8 / 33
Outline
Modeling
Examples of functionsFunctions expressed by formulasFunctions described numericallyFunctions described graphicallyFunctions described verbally
Properties of functionsMonotonicitySymmetry
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 9 / 33
Functions expressed by formulas
Any expression in a single variable x defines a function. In this case, thedomain is understood to be the largest set of x which after substitution,give a real number.
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Formula function example
Example
Let f (x) =x + 1
x − 2. Find the domain and range of f .
Solution
The denominator is zero when x = 2, so the domain is all real numbersexcept 2. As for the range, we can solve
y =x + 1
x − 2=⇒ x =
2y + 1
y − 1
So as long as y 6= 1, there is an x associated to y. Therefore
domain(f ) = { x | x 6= 2 }range(f ) = { y | y 6= 1 }
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
Formula function example
Example
Let f (x) =x + 1
x − 2. Find the domain and range of f .
Solution
The denominator is zero when x = 2, so the domain is all real numbersexcept 2.
As for the range, we can solve
y =x + 1
x − 2=⇒ x =
2y + 1
y − 1
So as long as y 6= 1, there is an x associated to y. Therefore
domain(f ) = { x | x 6= 2 }range(f ) = { y | y 6= 1 }
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
Formula function example
Example
Let f (x) =x + 1
x − 2. Find the domain and range of f .
Solution
The denominator is zero when x = 2, so the domain is all real numbersexcept 2. As for the range, we can solve
y =x + 1
x − 2=⇒ x =
2y + 1
y − 1
So as long as y 6= 1, there is an x associated to y.
Therefore
domain(f ) = { x | x 6= 2 }range(f ) = { y | y 6= 1 }
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
Formula function example
Example
Let f (x) =x + 1
x − 2. Find the domain and range of f .
Solution
The denominator is zero when x = 2, so the domain is all real numbersexcept 2. As for the range, we can solve
y =x + 1
x − 2=⇒ x =
2y + 1
y − 1
So as long as y 6= 1, there is an x associated to y. Therefore
domain(f ) = { x | x 6= 2 }range(f ) = { y | y 6= 1 }
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
How did you get that?
start y =x + 1
x − 2
cross-multiply y(x − 2) = x + 1
distribute xy − 2y = x + 1
collect x terms xy − x = 2y + 1
factor x(y − 1) = 2y + 1
divide x =2y + 1
y − 1
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 12 / 33
No-no’s for expressions
I Cannot have zero in thedenominator of anexpression
I Cannot have a negativenumber under an even root(e.g., square root)
I Cannot have the logarithmof a negative number
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 13 / 33
Piecewise-defined functions
Example
Let
f (x) =
{x2 0 ≤ x ≤ 1;
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.
0 1 2
1
2
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 14 / 33
Piecewise-defined functions
Example
Let
f (x) =
{x2 0 ≤ x ≤ 1;
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.
0 1 2
1
2
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 14 / 33
Functions described numerically
We can just describe a function by a table of values, or a diagram.
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Example
Is this a function? If so, what is the range?
x f (x)
1 42 53 6
1
2
3
4
5
6
Yes, the range is {4, 5, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
Example
Is this a function? If so, what is the range?
x f (x)
1 42 53 6
1
2
3
4
5
6
Yes, the range is {4, 5, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
Example
Is this a function? If so, what is the range?
x f (x)
1 42 53 6
1
2
3
4
5
6
Yes, the range is {4, 5, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
Example
Is this a function? If so, what is the range?
x f (x)
1 42 43 6
1
2
3
4
5
6
Yes, the range is {4, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
Example
Is this a function? If so, what is the range?
x f (x)
1 42 43 6
1
2
3
4
5
6
Yes, the range is {4, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
Example
Is this a function? If so, what is the range?
x f (x)
1 42 43 6
1
2
3
4
5
6
Yes, the range is {4, 6}.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
Example
How about this one?
x f (x)
1 41 53 6
1
2
3
4
5
6
No, that one’s not “deterministic.”
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
Example
How about this one?
x f (x)
1 41 53 6
1
2
3
4
5
6
No, that one’s not “deterministic.”
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
Example
How about this one?
x f (x)
1 41 53 6
1
2
3
4
5
6
No, that one’s not “deterministic.”
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
An ideal function
I Domain is the buttons
I Range is the kinds of sodathat come out
I You can press more than onebutton to get some brands
I But each button will onlygive one brand
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
An ideal function
I Domain is the buttons
I Range is the kinds of sodathat come out
I You can press more than onebutton to get some brands
I But each button will onlygive one brand
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
An ideal function
I Domain is the buttons
I Range is the kinds of sodathat come out
I You can press more than onebutton to get some brands
I But each button will onlygive one brand
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
An ideal function
I Domain is the buttons
I Range is the kinds of sodathat come out
I You can press more than onebutton to get some brands
I But each button will onlygive one brand
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
An ideal function
I Domain is the buttons
I Range is the kinds of sodathat come out
I You can press more than onebutton to get some brands
I But each button will onlygive one brand
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
Why numerical functions matter
In science, functions are often defined by data. Or, we observe data andassume that it’s close to some nice continuous function.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 20 / 33
Numerical Function Example
Here is the temperature in Boise, Idaho measured in 15-minute intervalsover the period August 22–29, 2008.
8/22 8/23 8/24 8/25 8/26 8/27 8/28 8/29
10
20
30
40
50
60
70
80
90
100
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 21 / 33
Functions described graphically
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.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 22 / 33
Functions described graphically
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.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 22 / 33
Functions described verbally
Oftentimes our functions come out of nature and have verbal descriptions:
I The temperature T (t) in this room at time t.
I The elevation h(θ) of the point on the equator at longitude θ.
I The utility u(x) I derive by consuming x burritos.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 23 / 33
Outline
Modeling
Examples of functionsFunctions expressed by formulasFunctions described numericallyFunctions described graphicallyFunctions described verbally
Properties of functionsMonotonicitySymmetry
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 24 / 33
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
0.5
$0 $52,115 $100K
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 25 / 33
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
0.5
$0 $52,115 $100K
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 25 / 33
Monotonicity
Definition
I A function f is decreasing if f (x1) > f (x2) whenever x1 < x2 for anytwo points x1 and x2 in the domain of f .
I A function f is increasing if f (x1) < f (x2) whenever x1 < x2 for anytwo points x1 and x2 in the domain of f .
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 26 / 33
Examples
Example
Going back to the burrito function, would you call it increasing?
Example
Obviously, the temperature in Boise is neither increasing nor decreasing.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 27 / 33
Examples
Example
Going back to the burrito function, would you call it increasing?
Example
Obviously, the temperature in Boise is neither increasing nor decreasing.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 27 / 33
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.
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Possible Intensity Graph
x
y = I (x)
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Possible Gravity Graph
x
y = F (x)
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Definitions
Definition
I A function f is called even if f (−x) = f (x) for all x in the domain off .
I A function f is called odd if f (−x) = −f (x) for all x in the domainof f .
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 31 / 33
Examples
I Even: constants, even powers, cosine
I Odd: odd powers, sine, tangent
I Neither: exp, log
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 32 / 33
Summary
I The fundamental unit of investigation in calculus is the function.
I Functions can have many representations
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