Transcript One to One Functions and Their Inverses

Section 2.8
One-to-One Functions and Their Inverses
What is a Function?
 Relationship between
inputs (domain) and
outputs (range) such
that each input produces
only one output
 Passes the vertical line
test
 OK for outputs to be
shared
One-to-One Functions
 A function is a one-to-one function if no outputs
are shared (each y-value corresponds to only one
x-value)
 Formal definition: If f(x1) = f(x2), then x1 = x2
 Passes the horizontal line test
One-to-One Functions
Horizontal Line Test
A function is oneto-one if and only
if each horizontal
line intersects the
graph of the
function in at
most one point.
y
y
y = f (x)
y = f (x)
(a, f (a))
a
(b, f (b))
b
(a, f (a))
x
a
x
Increasing and Decreasing Functions
y
x
y
y
(a) An increasing function
is always one-to-one
x
(b) A decreasing function
is always one-to-one
x
(c) A one-to-one function doesn’t
have to be increasing or
decreasing
Example 1:
Are the following functions one-to-one?
{(1, 1), (2, 4), (3, 9), (4, 16), (2, 4)}
Yes, each y-value corresponds to a unique x-value.
{(-2, 4), (-1, 1), (0, 0), (1, 1)}
No, there are 2 y-values of 1, which correspond
to different x-values.
This is a modified slide from the Prentice Hall Website.
What is an Inverse Function?
5
5
f(17)
g(5)
Function
Machine
Function
17
Machine
What is an Inverse Function? (cont’d)
x
x
g( )
Function
Machine
f [g(x) ]
gFunction
(Machine
x)
Inverse Function
 An Inverse Function is a new function that maps y-
values (outputs) to their corresponding x-values
(inputs).
 Notation for an inverse function is: f -1(x)
 For a function to have an inverse, it must pass the
horizontal-line test (i.e., only one-to-one functions
have an inverse function!)
 Why do we need a new function? Sometimes we
have the y-value for a function and we want to know
what x-value caused that y-value.
Formal definition of an Inverse Function

Let f be a function denoted by y = f(x). The
inverse of f, denoted by f -1(x), is a function
such that:


(f -1 ○ f )(x) = f -1[ f (x) ] = x for each x in the
domain of f, and
(f ○ f -1 )(x) = f [ f -1(x) ] = x for each x in the
domain of f -1.
Example 2:
3x  5
5
Verify that f ( x) 
and g ( x) 
are inverses.
x
x 3
5 

f ( g ( x ))  f 

 x  3
 5 
3
5

x 3
x 3



x 3
 5 
 x 3


 5 
3
5

x 3


 5 
 x 3


15  5 x  3 5x

x

5
5
This is a modified slide from the Prentice Hall Website.
Example 2 (cont’d):
3x  5
5
Verify that f ( x) 
and g ( x) 
are inverses.
x
x 3
5
3x  5

g ( f ( x ))  g 
 
 x   3x  5   3


x


5x
5
x

 
x
 3x  5 
3x  5  3x

3
 x 


5x
 x
5
Yes, these functions are inverses.
This is a modified slide from the Prentice Hall Website.
How do we find an Inverse Function?

If a function f is one-to-one, its inverse
can generally be found as follows:
1. Replace f(x) with y.
2. Swap x and y and then solve for y.
-1(x).
3. Replace y with f
Example 3:
5
Find the inverse of f ( x) 
, x3
x 3
5
y
x3
5
x
y3
xy  3x  5
xy  3x  5
3x  5
y
x
Replace f(x) with y
Swap x and y.
Solve for y.
3x  5
f ( x) 
x
1
This is a slide from the Prentice Hall Website.
What does an Inverse Function look like?
 Remember, to find an inverse function we just
interchanged x and y.
 Geometric interpretation: graph will be symmetrical
about the line y = x.
Domain of f
Range of f
f
f
Range of f 1
1
Domain of f 1
 The domain of the original function becomes the range of the
inverse function.
 The range of the original function becomes the domain of the
inverse function.
This is a slide from the Prentice Hall Website.