Composite Functions; One-to- one Functions
A composite function f ο g (read as “f composed
with g”) is defined by
( f ο g)(x) = f (g(x)).
The domain of f ο g is the set of all real x in the
domain of g for which g ( x) is in the domain of f .
Example: Show a diagram for the composite function
( f ο g)(x) = f (g(x))
Similarly we define: (g ο f )(x) = g( f (x))
Example:Let f (x) = sqrt(x − 5) and g(x) = x2 − 2. Find:
(a) ( f ο g)(4) =
(b) (g ο f )(9) =
(c) Find the composite functions and their domains
(g ο f )(x) =
Domain:
( f ο g)(x) =
Domain:
Example: Using the tables, find ( f ο g)(2) and (g
ο f )(1).
What is the value of ( f ο g)(0)?
Example: Find functions f and g such that f
ο g = h if
Example: An oil spill in the ocean as sumes a
circular
shape with an expanding radius r given by r = sqrt(t +1),
where t is the number of minutes after the measurements
are started and r is measured in meters.
(a) Find a formula that gives the area A of the circular
region as a function of time t.
(b) What is the area at the beginning? (t = )
(c) What is the area 3 minutes later? (t = )
Inverse Relations and Functions
Recall: A relation is a set of all ordered
pairs (x, y) where
x is an element from the domain of the relation and y is
the corresponding element from the range.
The inverse relation we defined as the set of all
ordered
pairs ( y, x).
Example: Find the inverse of the fol lowing
relations .
Which of the relations are functions? De termine whether
the inverse relations are functions.
{(−2,2),(−1,1),(0,0),(1,1),(2,2)}
{(−2,8),(−1,1),(0,0),(1,−1),(2,−8)}
Note: Not for every function the inverse relation
is a
function.
The inverse of a function is a function itself if and only
if
for each y in the range there is only one x in the domain.
In other words, no two ordered pairs have the same
second coordinates , that is, no horizontal line intersects
the graph in more than one point .
The functions for which the inverses are also functions
are called one-to-one.
Horizontal Line Test
If each horizontal line intersects the graph of a
function f in at most one point, then f is one-to-one.
Analytic Definition:
A function f is one-to-one if and only if, for
each a
and b in the domain of f ,
f (a) = f (b) ==> a = b
or
a ≠ b ==> f (a) ≠ f (b).
Example: Determine by using the analytic definition
which of the following functions is/are one-to-one:
f (x) = 3x − 2
f (x) = |x|
Example: Use the Horizontal Line Test to
determine
whether the function is one-to-one.
Note: A function, which is increasing/decreasing on an
interval I, is one-to-one on I.
Note: A quadratic function y = a(x − h)2 + k (a ≠ 0) is
not one-to-one, but when considered on the restricted
domain, for example, on interval [h,+∞), it is one-to-one.
Example: Determine whether the function
is one-to-one.
