A function from a set A to a set B is a relation
that assigns to each element x in the set A
exactly one element y in the set B.
set A = domain = the set of inputs
Sometimes the domain is not specified, if
the function is defined by an algebraic
expression the implied domain is the set of
numbers when the expression is defined.
range = the set of outputs (is included in set B)
  (1)
Verbally – a sentence
  (2) Numerically – a table or list of ordered
pairs
  (3) Graphically – points on the
coordinate plane
  (4) Algebraically – an equation
  KEY:
describes how the input is related to
the output (A rule to relate TWO sets)
  Each
element in set A must be matched
with an element in set B
  An element in set A cannot be matched
with two DIFFERENT elements in set B
  Two (or more) elements in set A may be
matched with the same element in set B
  Some elements in set B may not be
matched with any element in set A.
  Input
›  x, t, k, q, y, … (any variable)
  Output
›  f(x), g(t), h(k), m(q), p(y),…
  Equation
›  f(x)=1-x2, f(t)=1-t2, g(t)=(t+4)(t), g(y)=(y+4)(y)
If a function is given by: A=f(s)
  f
is the “name” of the function
›  f(s) is the value of the function at s
  A
is the dependent variable
  s is the independent variable
!
A={1,2,3,4,5} and
B{0,10,20,30,40,50,60}
{(1,30), (5,10), (4,50), (3,10)}
"!
#!
$!
%!
x −3= y
2y = x −1
2
2
x + y = 10
2
f (x) = x + 8 + 2
f(-8)
f(x-8)
f(1)
€
⎧9 − x 2 , x < 3
g(x) = ⎨
⎩ x − 3, x ≥ 3
  Fill
in the table
€
x g(x)
1
2
3
4
5
3
f (t) = t + 4
10
h(x) = 2
x − 2x
y
+
6
€ =
g(y)
6+ y
2
f (x) = 4 x − 2x
  Find
(h≠0)
€
f (x + h) − f (x)
h
  Sum
(f + g)(x) = f (x) + g(x)
  Difference
(f − g)(x) = f (x) − g(x)
  Product
(f g)(x) = f (x)g(x)
  Quotient
f
( g )(x)
  Composition
(f ◦ g)(x) = f (g(x))
=
f (x)
g(x)
Ex. 1: For f (x) =
Find
(a) (f + g)(x)
(b) (f g)(x)
(c) ( fg )(x)
(d) (f − g)(x)
√
x2
− 4 and g(x) =
x2
x2 +1 ,
Ex. 2: For f (x) =
√
x2 − 4 and g(x) =
Find
(a) (f ◦ g) and the implied domain
(b) g(f (x)) and the implied domain
x2
x2 +1 ,
Ex. 3: For f (x) = x3 − 1 and g(x) = 2x + 5,
Find
(a)
f
( g )(0)
(b) (f g)(2) + g(4)
(c) f (g(0))
Ex. 4: For f (x) = 3x + 5, find (f ◦ f )(x),
and the implied domain.
4
Ex. 5: Given h(x) = (5x+1)
2 , find two functions
f and g such that (f ◦ g)(x) = h(x).
Definition
If f(f -1(x)) = f -1(f(x)) = x , then f(x) and f -1(x) are
inverse functions.
CAUTION: f -1(x) is read “f inverse of x” the
“-1” indicates the inverse, it’s NOT an
exponent.
one-to-one function:
A function is one-to-one if each output
has exactly one input.
Note: A one-to-one function has an
inverse!
Two methods for finding an inverse:
1.  Graphically
2.  Algebraically
Ex. 1: Given f (x), find the inverse function f −1 (x)
(a) f (x) =
(b) f (x) =
√
3
x5 −1
3
x+2+1
Ex. 2: Show that f (x) =
and g(x) = 1−x
x , 0 < x ≤ 1,
are inverse functions.
1
1+x , x
≥ 0,
Ex. 3: Use f (x) = 3x + 4 and g(x) = x3
−1
to find (g ◦ f ) .