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S11MATH: General Mathematics

One to one Functions
and
Inverse Functions

One-to-One Functions
One-to-One Functions
Def
...


One-to-One Functions
Determining whether a set of ordered pairs (x,y) is 1-1
...
If it is a function, determine whether it is 1-1:
f  {(1, 2) (3,1) (1, 4)}
g  {(1, 2) (3, 4) (5, 2)}
h  {(1, 2) (3, 4) (5, 6)}

One-to-One Functions
Determining algebraically whether a function is oneto-one (1-1):
Recall: if f (x1 )  f (x2 ), then x1  x2
Determine whether the following functions are 1-1:
a
...
) g (x )  x2 1
c
...
) v (x)  16  x2

One-to-One Functions
Determining whether the graph of a given function
is one-to-one (1-1):
Horizontal Line Test
Def
...
Then the graph of a function is
considered to be one-to-one
...
if the function f is one-to-one containing a set of
ordered pairs (x,y), then the inverse of f is the set of
ordered pairs (y,x)
...
{(1, 2) (3, 4) (5, 6)}
2
...
{(1, 2) (2, 2) (3, 4)}

Inverse Functions
Inverse Function
Def
...

Then the inverse function of f (denoted by f 1 ) is the
function with domain R and range D defined by:

f 1 (b )  a if and only if f (a )  b
where a , b 



Inverse Functions
Properties of the inverse function by relating it to its
original function
...
"
Property 2: Inverse Composition Rule
( f 1 f )(x ) f 1 ( f (x ))  x ; for every x in the domain of f
(f

f 1 )(x ) f ( f 1 (x ))  x ; for every x in the domain of f -1

Inverse Functions
Verifying Inverse Functions
a) Verify that f

-1

x 4
(x ) 
2 is the inverse of f (x )  2x  4

b) Verify that g -1 (x )  ( x  1)3 is the inverse of g ( x )  3 x 1

Property 2: Inverse Composition Rule
( f 1 f )(x )  f 1 ( f (x ))  x ; for every x in the domain of f
(f

f 1 )(x )  f ( f 1 (x ))  x ; for every x in the domain of f -1

Inverse Functions
Verifying Inverse Functions with Domain Restrictions
a) Verify that f -1 (x )  x 2 for x  0 is the inverse of f (x )  x
b) Verify that h -1 (x )  x 2  7 for x  0 is the inverse of h (x )  x  7

Property 2: Inverse Composition Rule
( f 1 f )(x )  f 1 ( f (x ))  x ; for every x in the domain of f
(f

f 1 )(x )  f ( f 1 (x ))  x ; for every x in the domain of f -1

Inverse Functions
Algorithm for Finding the Inverse Function
Note: Verify first whether the function is 1-1 prior
to finding an inverse
...
e
...

Step 1: Let y  f (x)
Step 2: Interchange x and y
Step 3: Solve y in terms of x
Step 4: Let y  f -1 (x)

Inverse Functions
Find the inverse of the following functions
below then state their domain and range
...
These ordered pairs (x,y) and (y,x) is
symmetric with respect to the line y = x



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