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Chapter 0
Functions
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 78
Chapter Outline

Functions and Their Graphs

Some Important Functions

The Algebra of Functions

Zeros of Functions – The Quadratic Formula and Factoring

Exponents and Power Functions

Functions and Graphs in Applications
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 78
§ 0.1
Functions and Their Graphs
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 78
Section Outline

Rational and Irrational Numbers

The Number Line

Open and Closed Intervals

Applications of Functions

Domain of a Function

Graphs of Functions

The Vertical Line Test

Graphing Calculators

Graphs of Equations
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 78
Rational & Irrational Numbers
Definition
Rational Number: A number
that may be written as a finite
or infinite repeating decimal,
in other words, a number that
can be written in the form
m/n such that m, n are
integers
Irrational Number: A
number that has an infinite
decimal representation whose
digits form no repeating
pattern
Example

2
 0.285714
7
3  1.73205
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 78
The Number Line
The Number Line
A geometric representation of the real numbers is shown
below.

-6
-5
-4
-3
-2
-1
2
7
0
3
1
2
3
4
5
6
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 78
Open & Closed Intervals
Definition
Open Interval: The set of
numbers that lie between
two given endpoints, not
including the endpoints
themselves
Closed Interval: The set of
numbers that lie between
two given endpoints,
including the endpoints
themselves
Example
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
2
3
4
5
6
4, 
x4
-6
-5
-4
-3
-2
-1
0
1
[-1, 4]
1  x  4
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 78
Functions in Application
EXAMPLE
(Response to a Muscle) When a solution of acetylcholine is introduced into the heart muscle of
a frog, it diminishes the force with which the muscle contracts. The data from experiments of
the biologist A. J. Clark are closely approximated by a function of the form
Rx  
100 x
b x
where x is the concentration of acetylcholine (in appropriate units), b is a positive constant that
depends on the particular frog, and R(x) is the response of the muscle to the acetylcholine,
expressed as a percentage of the maximum possible effect of the drug.
(a) Suppose that b = 20. Find the response of the muscle when x = 60.
(b) Determine the value of b if R(50) = 60 – that is, if a concentration of x = 50 units produces a
60% response.
SOLUTION
(a)
Rx  
100 x
b x
This is the given function.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 78
Functions in Application
CONTINUED
R60 
100  60
20  60
Replace b with 20 and x with 60.
R60  
6000
80
Simplify the numerator and
denominator.
R60  75
Divide.
Therefore, when b = 20 and x = 60, R (x) = 75%.
(b)
Rx  
100 x
b x
This is the given function.
R50 
100  50
b  50
Replace x with 50.
60 
100  50
b  50
Replace R(50) with 60.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 78
Functions in Application
CONTINUED
60 
b  50 60 
5000
b  50
Simplify the numerator.
5000
 b  50
b  50
Multiply both sides by b + 50 and cancel.
60b  3000  5000
60b  2000
b  33.3
Distribute on the left side.
Subtract 3000 from both sides.
Divide both sides by 60.
Therefore, when R (50) = 60, b = 33.3.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 78
Functions
EXAMPLE
If f x   x 2  4 x  3 , find f (a - 2).
SOLUTION
f x   x 2  4 x  3
This is the given function.
f a  2  a  2  4a  2  3
2


Replace each occurrence of x with a – 2.
f a  2  a 2  4a  4  4a  2  3
Evaluate (a – 2)2 = a2 – 4a + 4.
f a  2  a 2  4a  4  4a  8  3
Remove parentheses and distribute.
f a  2  a 2  1
Combine like terms.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 78
Domain
Definition
Domain of a Function: The
set of acceptable values for
the variable x.
Example
The domain of the function
f x  
is
1
3 x
3 x  0
3 x
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 78
Graphs of Functions
Definition
Example
Graph of a Function: The set
of all points (x, f (x)) where x is
the domain of f (x). Generally,
this forms a curve in the xyplane.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 78
The Vertical Line Test
Definition
Example
Vertical Line Test: A curve in
the xy-plane is the graph of a
function if and only if each
vertical line cuts or touches the
curve at no more than one
point.
Although the red line intersects
the graph no more than once
(not at all in this case), there
does exist a line (the yellow
line) that intersects the graph
more than once. Therefore,
this is not the graph of a
function.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 14 of 78
Graphing Calculators
Graphing Using a Graphing Calculator
Step
Display
1) Enter the expression
for the function.
2) Enter the specifications
for the viewing window.
3) Display the graph.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 15 of 78
Graphs of Equations
EXAMPLE


1
2
Is the point (3, 12) on the graph of the function f x    x  x  2 ?
SOLUTION
1

f  x    x   x  2 
2

This is the given function.
1

f 3   3  3  2 
2

Replace x with 3.
1

12   3  3  2
2

Replace f (3) with 12.
12  2.55
Simplify.
12  12.5
false
Multiply.
Since replacing x with 3 and f (x) with 12 did not yield a true statement in the original function, we
conclude that the point (3, 12) is not on the graph of the function.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 16 of 78