Transcript 9.5 Functions

9.5 Functions
CORD Math
Mrs. Spitz
Fall 2006
Objectives
• Determine whether a given relation is
a function, and
• Calculate functional values for a
given function
Assignment
• Pgs. 377-378 #5-48 all
Ex. 1: Is {(5, -2), (3, 2), (4, -1),
(-2, 2)} a function?
• Since each element of the domain is
paired with exactly one element of
the range, this relation is a function.
Ex. 2: Which mapping
represents a function?
X
Y
2
3
5
-1
4
3
0
2
1
The mapping in
this mapping
represents a
function since, for
each element of
the domain, there
is only one
corresponding
element in the
range.
Ex. 3: Which mapping
represents a function?
X
Y
3
0
4
3
6
4
-2
7
2
The mapping in
this relation does
not represent a
function since the
element 4 in the
domain maps to
two elements, -3
and 4, in the
range.
Ex. 3: Is the relation represented by the
equation x + 2y = 8 a function?
• Substitute a value for x in the equation.
What is the corresponding value of y? Is
there more than one value for y? For
example, if x is 2, then y is 3 and that is
the only value of y that will satisfy the
equation. If you try other values of x, you
will see that there is always only one
corresponding value of y. Therefore, the
equation x + 2y = 8 represents a function.
Notes
• For equations like x + 2y = 8, it may
not be easy to determine whether
there is an element of the domain
that is paired with more than one
element of the range. Often it is
simpler to look at the graph of the
relation. Suppose you graph x + 2y
= 8.
First solve for y
x + 2y = 8
-x
-x
2y = 8 – x
2
2
y=8–x
2
Make a table of values and
graph the equation.
x
2
y
3
6
4
0
x + 2y = 8
4
2
8
0
5
10
-2
Now place your pencil at the left of the graph to represent a vertical
line. Slowly move the pencil to the right across the graph.
For each value of x, this vertical line passes through no more than
one point on the graph. This is true for EVERY function.
Vertical Line Test for a
Function
• If any vertical line passes through no
more than one point of the graph of a
relation, then the relation is a
function.
Ex. 4: Use the vertical line test to
determine if each relation is a function.
Okay be careful. Note the point!
Functional Notation
• Equations that represent functions can be
written in a form called functional notation.
The equation y = 2x + 1 can be written in
the form f(x) = 2x + 1. The symbol f(x) is
read “f of x” and represents the value in
the range of the function that corresponds
to the value of x in the domain. For
example f(3) is the element of the range
that corresponds to the element x = 3 in
the domain. We say f(3) is the functional
value of f for x = 3.
Functional Notation
• Letters other than f are also used for
names of functions.
• The ordered pair (3, f(3)) is a solution
of the function f in the previous slide.
• You can determine a functional value
by substituting the given value for x
into the equation. For example, if
f(x) = 2x +1 and x = 3, then f(3) =
2(3) + 1 or 7.
Ex. 5 If f(x) = 3x – 7, find each
of the following:
A. Find f(2)
f(2) = 3 (2) – 7
=6–7
= -1
B. Find f(5)
f(5) = 3 (5) – 7
= 15 – 7
=8
B. Find f(-3)
f(-3) = 3 (-3) – 7
= -9 – 7
= -16
Ex. 6 If g(x) = x2 – 2x + 1, find
each of the following:
A. Find g(6a)
g(6a) = (6a)2 – 2(6a) + 1
= 36a2 – 12a + 1
B. Find 6[g(a)]
6[g(a)] = 6[a2 – 2(a) + 1
= 6a2 – 12a + 6