Lecture 1 of MATH 138 -admin stuff -representing functions

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Math 138
Section 003
Professor Brown
Fall 2014

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Prof Brown Contact Info
ronbrown@njit.edu (best)
973-642-4096 (I don’t check that often)
Office Hours:
Tuesday 10 to 1
Wednesday 11:30 to 12:30
CULM 212A (back of the adjunct office)

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Course Information
Start with Moodle page if you have any questions
Class Expectations
Academic Integrity Policy
Attendance Policy
Quiz Policy
Exam Policy

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Grading
Quizzes 15%
Midterm 1 25%
Midterm 2 25%
Final 35%

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Resources
Textbook
My office hours
Math Learning Center (CULM 214)
Internet (Kahn Academy, etc)

6. Four Ways to Represent 1.1 a Function

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What is a Function?
Functions arise whenever one quantity depends on another.
We usually consider functions for which the sets D and E
are sets of real numbers. The set D is called the domain of
the function.
The number f (x) is the value of f at x and is read “f of x.”
The range of f is the set of all possible values of f (x) as x
varies throughout the domain.
A symbol that represents an arbitrary number in the domain
of a function f is called an independent variable.

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What is a Function?
It’s helpful to think of a function as a machine
Domain
Independent Variable
Range
Dependent Variable
or y
Key Idea – each input has only one output

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Representations of Functions
There are four possible ways to represent a function:
verbally (by a description in words)
numerically (by a table of values)
algebraically (by an explicit formula)
visually (by a graph)

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Verbal Representation
• Revenue is $10 for every unit sold
• Force required to stretch/compress a spring is
proportional to the distance spring is
stretched/compressed
• Voltage across a capacitor decays exponentially
with a time constant RC

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Numerical Representation
The human population of the world P depends on the
time t. The table gives estimates of the world
population P(t) at time t, for certain years. For instance,
P(1950) » 2,560,000,000
But for each value of the time t
there is a corresponding value
of P, and we say that P is a
function of t.

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Algebraic Representation
The area of a circle A = p r
2
Height of projectile: h(t) = -16t2+ vt + h0

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Graphical Representation
The graph of f also allows us to picture the domain of f on
the x-axis and its range on the y-axis as in Figure 5.
Figure 5

14. Example 1 – Reading Information from a Graph
The graph of a function f is shown in Figure 6.
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Figure 6
The notation for intervals is given in Appendix A.
(a) Find the values of f (1) and f (5).
(b) What are the domain and range of f ?

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Example 1 – Solution

16. Vertical Line Test
The graph of a function is a curve in the xy-plane. But the question
arises: Which curves in the xy-plane are graphs of functions? This
is answered by the following test.
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Representations of Functions
The reason for the truth of the Vertical Line Test can be
seen in Figure 13.
Figure 13
If each vertical line x = a intersects a curve only once, at
(a, b), then exactly one functional value is defined by
f (a) = b. But if a line x = a intersects the curve twice, at (a, b)
and (a, c), then the curve can’t represent a function because
a function can’t assign two different values to a.

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Vertical Line Test
Draw a graph that fails the vertical line test at x=3
Draw a graph that passes the vertical line test.

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Finding Domains
• Sometimes explicitly given
• Sometimes implied – word problem describing
area of a circle as a function of radius, implies
radius is positive number
• Exclude values that would “break” the function:
• No division by zero
• No even roots of negative numbers
• No logs of non-positive numbers

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Finding Domains - Examples

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Finding Domains - Examples

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Finding Domains - Examples

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Finding Domains - Examples