Station 1: What is a function?
Information:
Relation: correspondence between two sets
 x corresponds to y AND y depends on x
 xy
 x, y 
Function: a relation that associates each element of one set x with exactly one element of set y
(x cannot repeat; vertical line test)
Domain: set of x-coordinates
Range: set of y-coordinates
Domain
(set X)
Function Notation: f x  read “f of x”
Range
(set Y)
(not f times x)
f is the notation for the function which associates the domain element x to the range
element, f(x)
Independent Variable:
Dependent Variable:
x
f(x) or y
Implicit form of a function: When the function is given as an equation in terms of x and y, such as
2x  y  6 .
Explicit form of a function: when the function is given as y in terms of x, such as f x   6  2x .
Solve the following problems using the information on functions.
Determine which relations are functions.
A)
Bob
Fran
Susan
Rick
Apr 7
May 4
Sept 5
B){ (2, 5), (-2, 5), (2, 7)}
Determine if the equation is a function. Sketch a graph if necessary.
a) y = 5x – 1
b) (x + 4)2 = -8 (y + 2)
c) y2 + 2x = 3
For the function f defined by f(x) = 3x2 + 1, evaluate the following.
a) f(2)
b) f(-5)
c) f( ½ )
d) f x  2
e)
f  x  h   f x 
; h0
h
Station 2: Finding domain of a function
Information:
The domain of a function will be all reals, except for the values of x that violate a rule of
math (zero in the bottom of a fraction, or a negative number under a square root)
Examples: Find the domain of each function; write your answer using interval notation
A. y  3x 2  5x  1
Because there is no restriction on what number can be substituted for the x, the domain is the set
of all real numbers.
Using interval notation, this is written (, )
B. y  3 x  12
Because we cannot have a negative number under the square root symbol, we must begin by writing
an inequality to show the radicand (expression under radical sign) is non-negative, and then solve
the inequality.
3 x  12  0
3 x  12
x4
Using interval notation, this is written
C. y 
[4,  )
x4
x  6x  5
2
Because we cannot have zero for a denominator, we must begin with an inequality to show this, and
then solve the inequality.
x2  6 x  5  0
( x  1)( x  5)  0
x  1  0 and x  5  0
x  1 and
x5
Using interval notation, this is written
(, 1), (1,5), (5,  )
Domain
If the domain is not specified, then the domain is the largest set of real numbers for which f(x)
is a real number.
Find the domain of each function. Write the domain in interval notation.
a) f x   2 x  x 2
c) h t   3  2t
e) f ( x)  3x  7
b) gx  
d)
f)
2x
x 1
f ( x) 
h( x ) 
3x  6
x 2  25
3x  15
x6
Station 3: Operations on functions
(Add, subtract, multiply, divide)
Operations on Functions

Sum Function
 f  gx   f x   gx 

Difference Function
 f  gx   f x   gx 

Product Function
 f  gx   f x   gx 
(Domain of each of these consists of any x that are in the domain of f and g)

Quotient Function
f
f x 
 x  
gx 
 g
(Domain consists of any x in the domains of f and g, such that gx   0 )
Example: For the given functions f and g, f ( x)  5 x  7 and g ( x)  x 2  4 find the following and
determine the domain for each.
( f  g )( x)  f ( x)  g ( x )
 (5 x  7)  ( x 2  4)
 x 2  5 x  11
 g  f  ( x)  g ( x) 
Domain: (, )
f ( x)
 ( x 2  4)(5 x  7)
 5 x 3  7 x 2  20 x  28
g
g ( x)
  ( x) 
f ( x)
 f 
x2  4

5x  7
Domain: (, )
7 7 

Domain:  ,  ,  ,  
5 5 

Find the following and determine the domain of each:
a)  f  gx 
c)
 f  g  x 
f x  
f
x 
 g
b) 
d)
 f g  x 
2
x
and gx  
x 1
x 1
Station 4: Composition of functions
Given two functions f and g, the composite function, denoted by
(Read as “f composed with g” or “f of g”), is defined by
The domain of
g   ( f ( g ( x))
 f (2 x  3)
 (2 x  3) 2  3(2 x  3)  1
 4 x 2  12 x  9  6 x  9  1
 4 x 2  18 x  17
g
f   ( g ( f ( x))
g ( x 2  3 x  1)
 2 ( x 2  3 x  1)  3
 2x2  6x  2  3
 2x2  6x  1
g
g    f g  x   f  g  x   .
f  g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.
Suppose that f  x   x 2  3x  1 and g  x   2 x  3 .
f
f
f g
f  2   g ( f (2)
 g (22  3  2  1)
 g (9)
 29 3
 21
Suppose that f x   2 x and g x   3x 2  1 . Find the following:
A.)
f
B.) g  f 2
g  x 
C.)
Use the graphs of f and g to answer the following:
A.)  f  g 4
B.)
g  f 4
g(x)
f(x)
C.) g  f 2
D.)
 f  g 2
E.)  f  f  3
F.)
g  g 0
Find  f  g  x  if f  x  
3
;
x 1
g x  
2
and give its domain.
x
 f  f  3