Download Functions—3.1 & 3.2

Math 2
Precalculus Algebra
Name_____________________________
Functions—3.1 & 3.2
Correspondence/Relation
A correspondence is a relationship between two sets. A correspondence or one set mapping to another may be denoted by an arrow “”.
Examples of correspondences:
A set of students at Cabrillo  a set of student ID numbers
A set of chairs in the classroom  a set of Cabrillo students
A set of numbers denoted by x  a set of numbers denoted by y
Functions
A function, f, is a correspondence between two sets in which each element in the first set (x) is assigned to only one element in the
second set (y). In other words, for every input (x), there is only one corresponding output (y).
Are any of the above correspondences a function?
Set Notation of Functions
f
x
y
f
D 
→E
D
Domain
“first set”
“inputs”
argument
or
f :D→E
E
Range
“second set”
“outputs”
Domain
The domain of a function is the set of all x-values (elements in the first set of the correspondence).
Range
The range of a function is the set of all y-values (elements in the second set of the correspondence).
Function Notation
A function takes an input and gives an output.
A function takes an input = an output.
A function ( input ) = an output.
.f ( input) = output.
.f ( x ) = output.
Example:
A function takes an input and gives its square. In function notation:
f ( i n p u t ) = i n p u t2
or
f ( x ) = x2
Which of the following correspondences are functions? If the relation is a function, give its domain and range.
1.
Nicknames
Robert
Margaret
2.
Bob
Robby
Peggy
Megan
A number squared
0
3
-3
4
0
9
16
Over
3.
{ ( − 1 , 1 ) , ( 0 , − 3 ) , ( 3 , 4 ) , ( 4 , 5 ) , ( − 2 , − 2 )}
Ordered Pairs:
Function Notation
The value of f at the number x is denoted by f ( x ) . x is the independent variable snd y = f ( x ) is the dependent variable.
4.
For the function f defined by f ( x ) =
− x 2 + 7 x − 1 1 , evaluate;
b)
1
f  
2
f ( 2x )
c)
− f ( x)
d)
f ( −x )
e)
f ( x + 3)
f)
f ( x) + 3
a)
g)
5.
3
For the function g defined by g ( x ) =
b)
1
g 
2
g ( 2x )
c)
−g ( x )
d)
g ( −x )
e)
g ( x) + 2
a)
6.
f ( x + 3) − f ( x )
For the function g defined by g ( x ) =
g (a + h) − g (a)
a)
b)
c)
d)
e)
h
,
x2 + 4 x
, evaluate;
x+6
x2
, evaluate
x +1
if h ≠ 0
1
g 
a
1
g (a)
g
( a)
g (a)
Operations on Functions
If f ( x ) represents one function and g ( x ) represents a second function, and x is in the domain of both functions, then the following
operations on functions may be preformed:
Sum of functions:
Difference of functions:
Product of functions:
Quotient of functions:
( f + g )( x ) = f ( x ) + g ( x )
( f − g )( x ) = f ( x ) − g ( x )
( f ⋅ g )( x ) = f ( x ) ⋅ g ( x )
f ( x)
, provided that g ( x ) ≠ 0
( f / g )( x ) =
g ( x)
For the given functions f and g, find the following:
a)
(f
+ g )( x )
b)
(f
− g )( x )
c)
( f ⋅ g )( x )
e)
(f
+ g )( 3)
f)
(f
− g )( 4 )
g)
( f ⋅ g )( 2 )
7.
f ( x=
)
x −1 ;
g ( x) =
2
x
Find the difference quotient of f; that is, find
8.
f ( x) =
−3 x + 1
9.
f ( x) =
− x2 + 2 x + 3
10.
f ( x) =
 f 
d)   ( x )
g
 f 
h)   (1)
g
f ( x + h) − f ( x)
h
, h ≠ 0 , for each function. Be sure to simplify.
1
x+3
Finding the Domain
To find the domain of a function given the equation:
1. Assume that the domain is all real numbers.
2. Take out numbers (or intervals of numbers) for which the function is undefined:
a) Take out x’s that make a denominator zero. Set denominator = 0 to find numbers to omit from the domain.
b) Take out x’s that cause imaginary numbers (even roots of negative radicands). (Set radicand ≥ 0 to find x’s that work.
c) Take out x’s for which the argument of a logarithm is negative. (This will be covered in Chapter 6).
Find the domain of each function.
11.
f ( x=
) x3 + 1 0
12.
g (=
x)
13.
h ( x) =
14.
f ( x) =
x−5
5
x+2
1− x
15.
x + 2x − 1 5
f ( x=
) x2 − 2
16.
G ( x) =
2
x+4
x − 4x
3
Vertical Line Test
The graph of a set of points, ( x , y ) , in a coordinate plane is the graph of a function if every (and any) vertical line intersects the graph
in at most one point.
Explain why the graph is or is not the graph of a function.
17.
Use the given graph of the function f to answer parts (a)–(l):
a) Find f ( 0 ) and f ( − 2 ) ;
b) Is f ( − 1) positive or negative?
c)
For what values of x is f ( x ) = 0 ?
d) For what values of x is f ( x ) > 0 ?
18.
e)
f)
g)
h)
i)
j)
k)
What is the domain of f ?
What is the range of f ?
What are the x-intercepts?
What is the y-intercept?
How often does the line y = 1 intersect the graph?
How often does the line x = 2 intersect the graph?
For what values of x does f ( x ) = 1 ?
l)
For what values of x does f ( x ) = 3 ?
f ( x) =
x2 + 2
x+4
 3
Is the point  1,  on the graph of f ?
 5
b) If x = 0 , what is f ( x ) ? What point does this represent on the graph of f ?
a)
1
, what is x? What point(s) does this represent on the graph of f ?
2
d) What is the domain of f ?
e) List the x-intercepts, if any of the graph of f.
f) List the y-intercept, if any of the graph of f.
c)
If f ( x ) =