Transcript Section 1.2
1.2 Functions
Pre-Calculus
Introduction to Functions
1.
2.
Many everyday phenomena involve pairs of
quantities that are related to each other by some
rule of correspondence. The mathematical term for
such a rule of correspondence is a relation.
The simple interest I earned on an investment of
$1000 for 1 year is related to the annual interest
rate r by the formula I=1000r.
The area A of a circle is related to its radius r by the
formula A r 2
Function
Not all relations have simple math formulas.
People commonly match up NFL starting
quarterbacks with touchdown passes, and
the hours of day with temperature. In each of
theses cases, there is some relation that
matches each item from one set with exactly
one item from a different set.
This is called a function.
Definition of a Function
Function – from a set A to a set B
is a relation that assigns to each
element x in the set A exactly one
element y in the set B.
Domain – Set A (input)
Range – Set B (output)
Characteristics of a Function from Set
A to Set B
1.
2.
3.
4.
Each element of A must be matched with an
element of B.
Some elements of B may not be matched
with any element of A.
Two or more elements of A may be matched
with same element of B.
An element of A (the domain) cannot be
matched with two different elements of B.
Testing for Functions
Example 1: Decide whether the table represents y as a function
of x.
X
Y
-3
5
-1
-12
0
5
2
3
4
1
Testing for Functions Represented
Algebraically
Which of the equations
represent(s) y as a function
of x?
(a)
(b)
x y 1
2
x y 1
2
Function Notation
When an equation is used to represent a
function, it is convenient to name the function
so that is can be referenced easily. For
example, you know that the equation y = 1 x2 describes y as a function of x. Suppose
you give this function the name “f”.
Input (x) Output f(x)
Equation f(x) = 1 – x2
Evaluating a Function
(a)
(b)
(c)
Let g(x) = -x2 + 4x +1.
g(2)
g(t)
g(x + 2)
A Piecewise-Defined Function
Evaluate the function
when x = -1 and x =0.
x 1, x 0
f ( x)
x 1, x 0
2
Domain of a Function
The domain of a
function can be
described explicitly or it
can be implied by the
expression used to
define the function.
The implied domain is
the set of all real
numbers for which the
expression is defined.
Examples
(a)
1
f ( x) 2
x 4
(b)
f ( x) x
Finding the Domain of a Function
Find the domain of each
function.
(a) f: {(-3, 0), (-1, 4), (0,
2), (2, 2), (4, -1)}
(b) g(x) = -3x2 +4x +5
(c) h(x) = 1/(x+5)
(d) Volume of a sphere: V
(e)
k ( x) 4 3x
4 3
r
3
Cellular Communications Employees
The number N (in
thousands) of employees in
the cellular communications
industry in the U.S.
increased in a linear pattern
from 1998 to 2001. In 2002,
the number dropped, then
continued to increase
through 2004 in a different
linear pattern. These two
patterns can be
approximated by the
function
23.5t 53.6,8 t 11
N (t )
16.8t 10.4,12 t 14
Where t represents the
year, with t=8
corresponding to 1998.
Use this function to
approximate the
number of employees
for each year from 1998
to 2004.
Baseball
A baseball is hit at a point 3
feet above the ground at a
velocity of 100 feet per
second and an angle of 45
degrees. The path of the
baseball is given by the
function
f(x) = -0.0032x2 + x +3 where x
and f(x) are measured in
feet. Will the baseball clear
a 10-foot fence located 300
feet from home plate?
Homework
Page 24-29
2-10 even, 13-23 odd, 28-42 even, 53-61 odd,
79-82 all, 85