Transcript Section 1.3 Functions

Section 1.3 Functions
What you should learn
• How to determine whether relations
between two variables are functions
• How to use function notation and evaluate
two functions
• How to find the domains of functions
• How to use functions to model and solve
real-life problems
Relation
• A relation is a set of ordered pairs
of real numbers.
F = {(3, 2) (4, 1) (2, 4) (1, 3)}
• If I say (2, __ ) , can you fill in the
blank?
G = {(3, 3) (4, 1) (2, 1) (1, 3)}
• If I say (4, __ ) , can you fill in the
blank?
Domain
F = {(3, 2) (4, 1) (2, 4) (1, 3)}
• In a relation the set of all of the
values of the independent variable
is called the domain.
• What is the domain of F?
{3, 4, 2, 1}
• Does G = {(3, 3) (4, 1) (2, 1) (1, 3)}
have the same domain?
Range
G = {(3, 3) (4, 1) (2, 1) (1, 3)}
• In a relation the set of all of the
values of the dependent variable is
called the range.
• What is the range of G?
{3, 1}
• Does F = {(3, 2) (4, 1) (2, 4) (1, 3)}
have the same range?
(Domain, Range)
• Notice the alphabetical characteristic of
Domain and Range.
(x, y)
(a, b)
(abscissa, ordinate)
• Unfortunately (independent, dependent)
breaks the rule.
Function
• A function is a relation in which , for each
value of the first component there is
exactly one value of the second
component.
H = {(3, 2) (4, 1) (3, 4) (1, 3)}
K = {(2, 3) (4, 1) (3, 2) (1, 3)}
• H is not a function,but K is a function.
Definition of a Function (page 27)
• A function from set A to set B is a
relation that assigns to each element
x in the set A exactly one element y in
the set B.
• The set A is the domain (or set of
inputs) of the function f.
• The set B contains the range (or the
set of outputs)
Function Expressed as a
Mapping
Domain
A
Range
1
C
B
2
3
F=
{(A,1)
(C, 2)
(B, 3)}
Function Expressed as a
Mapping
Domain
Range
A
4
1
C
B
2
3
G=
{(A,1)
(C, 2)
(B, 3)
(A, 4)}
Since A goes to two ranges G is not a function.
Characteristics of a function from
Set A to Set B (page 27)
1. Each element in A must be matched with
an element in B.
2. Some elements in B may not be matched
with any element in A. (leftovers)
3. Two or more elements in A may be
matched with the same element in B.
4. An element in A (the domain) cannot be
matched with two different elements in B.
Four Ways to Represent a Function
1. Verbally by a sentence that describes how the
input variable is related to the output variable.
2. Numerically by a table or a list of ordered
pairs that matches input values with output
values
3. Graphically by points on a graph in a
coordinate plane in which the inputs are
represented on the horizontal axis and the
output values are represented by the vertical
axis.
4. Algebraically by an equation in two variables.
Testing for Functions Example 1a
Determine whether the relation represents y
as a function of x.
The input value x is the number of
representatives from a state, and the
output value y is the number of senators.
(x, 2)
• This is a constant function.
Testing for Functions Example 1b
Determine whether the
relation represents y
as a function of x.
Since x = 2 has two
outputs the table does
not describe a
function.
Input x
Output y
2
11
2
10
3
8
4
5
5
1
Testing for Functions Represented
Algebraically Example 2a
• Solve for y
• For each
value of x
there is only
one value
for y.
• So y is a
function of x.
x  y 1
2
y  x 1
2
Testing for Functions Represented
Algebraically Example 2b
• Solve for y
• For each
value of x
there are
two values
for y.
• So y is not a
function of x.
 x  y 1
2
y  x 1
2
y   x 1
Functional Notation
• y = F(x)
• F(x) read F of x
• It does not mean F × x
(multiplication)
Functional Notation
• Consider y = 2x + 5
• Suppose that you wanted to tell someone
to substitute in x = 3 into an equation.
• With functional notation y =
becomes f(x) = 2x + 5.
2x + 5
• And f(3) means substitute in 3
everyplace you see an x.
Example 3a Evaluating a Function
Find g(2)
g ( x)   x  4 x  1
2
g (2)  (2)  4(2)  1
g (2)  4  8  1
g (2)  5
2
Example 3b Evaluating a Function
Find g(t)
g ( x)   x  4 x  1
2
g (t )  (t )  4(t )  1
2
g (t )  t  4t  1
2
Example 3c Evaluating a Function
Find g(x+2)
g ( x)   x  4 x  1
2
g (t )  ( x  2)  4( x  2)  1
2
g (t )  ( x  4 x  4)  4 x  8  1
2
g (t )   x  4 x  4  4 x  8  1
2
g (t )   x  5
2
Example 4 A Piecewise-Defined
Function
 x  1, x  0
f ( x)  
 x  1, x  0
2
x
y
-2
5
-1
2
0
-1
1
0
2
1
The Domain of a Function
• The implied domain is the set of all real
numbers for which the expression is
defined.
• For what values of x is f(x) undefined?
x  2
1
f ( x)  2
x  4 {x | x  2}
The Domain of a Function
• The implied domain is the set of all real
numbers for which the expression is
defined.
• For what values of x is g(x) undefined?
g ( x)  x  8
x 8  0
{x | x  8}
Example 7 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°. The path of the ball is
given by the function
f ( x)  0.0032 x  x  3
2
• Will the baseball clear a10-foot fence
located 300 feet from home plate?
Example 9 Evaluating a Difference
Quotient
For
find
f ( x)  x  4 x  7
2
f ( x  h)  f ( x )
h
Homework
• Page 35
• 1 - 67 odd, 77, 79, 93