Transcript QUADRATIC FUNCTIONS

SECTION 1.1
FUNCTIONS
DEFINITION OF A
RELATION
A relation is a correspondence
between two sets. If x and y are two
elements in these sets and if a
relation exists between x and y, we
say that x corresponds to y or that y
depends upon x. The
correspondence can be written as an
ordered pair (x,y).
DEFINITION OF A
RELATION
Thus, a relation is simply a set of
ordered pairs or a table which relates
x and y values.
DEFINITION OF
FUNCTION
Let X and Y be two nonempty sets of
real numbers. A function from X into
Y is a rule or a correspondence that
associates with each element of X a
unique element of Y. This is a special
type of relation.
For every x, there is only one y!
DOMAIN AND RANGE
DEFINITION The set of all x values.
OF
DOMAIN
DEFINITION The set of all y values.
OF
Also called “functional
RANGE
values”.
THE FUNCTION AS A
“MAPPING”
x-values y-values
1
8
4
2
7
0
-3
-2
DOMAIN RANGE
Ordered Pairs:
(1 , 2)
(4 , 8)
(7, - 3)
(- 2, 0)
THE FUNCTION AS A
“MAPPING”
Consider 3 students whose names are
mapped to their letter grades on the
last History exam:
Jill
A
Frank
B
Sue
C
For each person in
the domain, there
can be only one
associated letter
grade in the range.
THE SQUARING
FUNCTION
-2
-1
0
0
1
1
4
2
3
9
Each element
in the domain
maps to its
square.
COUNTER-EXAMPLE:
4
1
2
5
3
Ordered Pairs:
(4, 1)
(4, 2)
(5, 3)
This is an example of a relation but
not a function.
THREE WAYS TO
REPRESENT A FUNCTION

NUMERICALLY - ordered pairs

SYMBOLICALLY - equation

GRAPHICALLY - picture
EXAMPLE
Determine whether the relation
represents a function:
(a) {(1,4),(2,5),(3,6),(4,7)}
EXAMPLE
Determine whether the relation
represents a function:
(a) {(1,4),(2,4),(3,5),(6,10)}
EXAMPLE
Determine whether the relation
represents a function:
(a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}
EXAMPLE
Determine whether the relation
represents a function:
(a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}
EVALUATING A FUNCTION
AT A GIVEN X-VALUE
x
f(x)
f(0) = 0
0
0
4
2
4
f(-2) = 4
-2
4
f(9) = 81
9
81
f(x) = x
f(2) =
2
Symbolically, the squaring function
can be represented as
y = x2
“FUNCTIONAL NOTATION”
f(x) = x 2
Read: “f of x equals x squared”
EVALUATING A FUNCTION
AT A GIVEN X-VALUE
For f(x) = 2x2 – 3x, find the
values of the following:
(a) f(3)
(b) f(x) + f(3)
(d) - f(x)
(e) f(x + 3)
(f) f(x  h)  f(x)
h
(c) f(-x)
FINDING VALUES OF A
FUNCTION ON A CALCULATOR
DO EXAMPLE 7
IMPLICIT FORM OF A
FUNCTION
When a function is defined by an
equation in x and y, we say that the
function is given implicitly. If it is
possible to solve the equation for y in
terms of x, then we write y = f(x) and
say that the function is given
explicitly. See examples on Pg 127.
DETERMINING WHETHER AN
EQUATION IS A FUNCTION
Determine if x2 + y2 = 1 is a function.
y   1 x
2
This means that for certain values of x,
there are two possible outcomes for y.
Thus, this is not a function!
Important Facts About
Functions:
1.
For each x in the domain of a function
f, there is one and only one image f(x)
in the range. For every x, there is only
one y.
2.
f is the symbol we use to denote the
function. It is symbolic of the
equation that we use to get from an x
in the domain to f(x) in the range.
f(x) is another name for y.
Important Facts About
Functions:
3.
If y = f(x) , then x is called the
independent variable or argument of
f, and y is called the dependent
variable or the value of f at x.
DOMAIN OF A FUNCTION
If a function is being described
symbolically and it comes with a
specific domain, that domain
should be expressly given.
Otherwise, the domain of the
function will be assumed to be the
“natural domain”.
EXAMPLE: f(x) =
2
x
Knowing the function of squaring a
number, we can determine that the
natural domain is all real numbers
because any real number can be
squared.
We can also look at a graph.
EXAMPLE: f(x) =
2
x
+ 5x
This is simply a modification of the
squaring function. Thus, we can
determine that the natural domain
is all real numbers.
We can also look at a graph.
EXAMPLE
Find the domain:
x  2
D: { x
x  2 }
3x
f(x) = 2
x 4
EXAMPLE
Find the domain:
4 – 3t  0
3t

4
t
3
4
f(t) = 4 - 3t
EXAMPLE
Find the domain : f(x) = x3 + x - 1
All real numbers
OPERATIONS ON
FUNCTIONS
Notation for four basic operations
on functions:
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f  g)(x) = f(x)  g(x)
(f / g)(x) = f(x) / g(x)
OPERATIONS ON
FUNCTIONS
Do Example 10
CONCLUSION OF SECTION 1.1