Composite Functions; Oneto one Functions ;
Inverse Functions
L13 Composite Functions; Onetoone Functions;
Inverse Functions
A composite function
(read as “f composed
with g”) is defined by
The domain of
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
Similarly we define:
Example: Let
and
g(x) = x^2 − 2. Find:
(c) Find the composite functions and their domains
Domain:
Domain:
Example: Using the tables, find
(1).
What is the value of
?
Example: Find functions f and g such that
if
Example: An oil spill in the ocean as sumes a circular
shape with an expanding radius r given by
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 Inverse Functions
Recall that 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.
Thus, 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 at more than one point.
The functions for which the inverses are also functions
are called onetoone.
Horizontal Line Test
If each horizontal line intersects the graph of a
function f in at most one point, then f is onetoone.
Example: Use the Horizontal Line Test to determine
whether the function is onetoone.
Note: A function which is increasing/decreasing on an
interval I is onetoone on I.
Note: A quadratic function y = a(x − h)^2 + k (a ≠ 0)
is not onetoone, but, when considered on the restricted
domain, for example, on interval [h,+∞), it is onetoone.
Inverse Functions
Remember, that the inverse of a function f is also a
function if and only if f is onetoone.
Let f be a onetoone function. Then g is the
inverse function of f if
for all x in the domain of g;
for all x in the domain of f.
If g is the inverse function of f, then we write g as
f^{1}(x) and read: “finverse”.
Example: Determine whether the following functions are
inverses of each other:
Cancellation Rules for Inverses
The inverse functions undo each other with respect to
their compositions:
f^{1}( f (x)) = x for all x in the domain of f
f(f^{1}( y)) = y for all y in the domain of f^{1}
Equivalent Form of the Cancellation Rules:
f (x) = y 
<> 
f^{1}(y) = x 
(x in domain of f) 

(y in the domain of f^{1}) 
Note on the Domains and Ranges of the Inverses:
Domain of f^{1} = Range of f
Range of f^{1} = Domain of f
Graphing Inverses:
If the graph of f is the set of points (x, y), then the
graph
of f^{1 }is the set of points ( y, x).
Since, points (x, y) and ( y, x) are symmetric with
respect to the line y = x
then
the graphs of f and f^{1} are symmetric with
respect to
the line y = x.
Example: Given the graph of
y = f (x). Draw the graph of
its inverse.
Finding the Inverse of a Onetoone Function f:
1. Write y = f (x).
2. Solve the equation for x: x = f^{1}( y)
3. Inter change x and y .
4. Give your answer in the form: y = f^{1}(x).
Note: Consider all restrictions on the variables .
Example: Find f^{1}(x) if it exists.
Finding the Inverse of a Domainrestricted Function:
Find the inverse of