MA111 CALCULUS I
Thursday, 1/26/12
 Today:
 Definition
of a function
 Ways
to describe and represent
functions
 Reading:
 Section 1.1
 Exercises:

Sec 1.1: 1-3, 5, 7-8, 13-14, 23-25, 27-28,
37-38, 41-45, 47, 51, 55,57
Thursday, 1/26/12, Slide #1
CQ #1
 Are
you (and your clicker) here?
 A.
Yes
No
 C. Not sure. Too early.
 B.
Thursday, 1/26/12, Slide #2
Key Concept: Function
A function is a rule which assigns to
every element of an input set (the
domain D) exactly one element of an
output set (the range E).
 You can think of a function as a
machine, taking in an input value and
putting out an output value.
 Or think of it as arrows pointing from
the domain to the range.

Thursday, 1/26/12, Slide #3
4 Ways to Represent a Function
 Algebraically
: Find a formula.
 Example?
 Visually
: Draw a graph.
 Example?
 Numerically
: Make a table of values.
 Example?
 Verbally
: Describe it in words.
 Example?
Thursday, 1/26/12, Slide #4
Two Examples from My Daily Life
1. My Prius’s gas
consumption table
 2. Weight Watcher’s online
point calculator (Macaroni
Grill Meatballs and Spaghetti)

Thursday, 1/26/12, Slide #5
Some Ideas about Functions of Real
Numbers

If not otherwise stated, the domain of a
function in calculus is the set of all real
numbers that can be put into the function.

Vertical Line Test for the graphs: A
graph in the plane is a function of the
horizontal variable if and only if each
vertical line intersects the graph at most
once.


Example?
Example?
Thursday, 1/26/12, Slide #6
CQ #2
 What
is the domain of this function:
1
f ( x) 
2x  4
 A.
All reals
All x > 2
 C. All x > 0
 D. All x > -2
 E. All x  -2
 B.
Thursday, 1/26/12, Slide #7
CQ #3
 Is
x2 + y2 = 9 a function?
 A.
Yes, because it passes
the Vertical Line Test.
 B. No, because it does not pass
the Vertical Line Test.
 C. y is a function of x , but x is not
a function of y .
Thursday, 1/26/12, Slide #8
Piecewise-Defined Functions
 Functions
defined by formula may be
defined by different formulas on
different parts of the domain.
 Example?
 Example:
http://tinyurl.com/3rtms
A
simple but important example is
the absolute value function |x|.

How do we define |x|?
Thursday, 1/26/12, Slide #9
CQ #4
 The
function f (x) = |x – 4| can be
“piecewise described” by:
x – 4 if x  - 4; x + 4 if x < - 4
x – 4 if x < 4; x + 4 if x  4
 C. x – 4 if x  4; -x + 4 if x < 4
 D. x if x  4; -x if x < 4
 E. You can search me
 A.
 B.
Thursday, 1/26/12, Slide #10
Mathematical Modeling
 This
means to try to capture realities of
some physical situation in terms of
functions.
 Often a labeled picture can be helpful.
 Example: A rectangle has an area of
40 square inches. Express the
perimeter of the rectangle as a function
of the length of one of its sides.
Thursday, 1/26/12, Slide #11
CQ #5
 Express
the area A of a circle as
a function of its diameter d.
A=d
 B. A = ¼  d 2
 C. A = ½ d 2
 D. A =  d 2
 E. A = ½  d
 A.
Thursday, 1/26/12, Slide #12