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A function and its inverse function can be described as the "DO" and the
"UNDO" functions. A function takes a starting value, performs some
operation on this value, and creates an output answer. The inverse
function takes the output answer, performs some operation on it, and
arrives back at the original function's starting value.
This "DO" and "UNDO"
process can be stated as a composition of functions. If functions
f and g are inverse functions,  . A function
composed with its inverse function yields the original starting
value. Think of them as "undoing" one another and leaving you right
where you started.
| So how do we find the inverse of a
function? |
Basically speaking, the process of finding an
inverse is simply the swapping of the x
and y coordinates. This newly formed inverse will be a
relation, but may not necessarily
be a function.
Consider this subtle difference in
terminology:
Definition: INVERSE OF A
FUNCTION: The relation formed when the independent
variable is exchanged with the dependent variable in a given
relation. (This inverse may NOT be a function.)
Definition: INVERSE FUNCTION: If the above mentioned
inverse of a function is itself a function, it is then called an inverse function.
 |
Remember:
The inverse of a function may not
always be a function!
The original function
must be a one-to-one function to guarantee that its inverse
will also be a function. |
| Definition: A function is a
one-to-one function if and only if each second
element corresponds to one and only one first element. (each x and y value is used only
once)
Use the horizontal line test to determine if a function is a
one-to-one function.
If ANY horizontal line
intersects your original function in ONLY ONE location, your
function will be a one-to-one function and its inverse will
also be a function.
The function y = 3x
+ 2, shown at the right, IS a one-to-one function and its
inverse will also be a function.
(Remember that the vertical line test is used to show
that a relation is a function.) |
| |
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Definition:
The inverse of a
function is the set of ordered pairs obtained by
interchanging the first and second elements of each pair in the
original function.
Should the inverse of function f (x) also be a
function, this inverse function is
denoted by f -1(x).
Note: If the original function
is a one-to-one function, the inverse will be a
function. |
[The notation f
-1(x) refers to "inverse function". It does
not algebraically mean 1/f (x).]
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If a function is
composed with its inverse function,
the result is the
starting value. Think of it as the function and the
inverse undoing one another when composed.
Consider the simple
function f (x) = {(1,2), (3,4), (5,6)}
and its
inverse f -1(x) = {(2,1), (4,3),
(6,5)}
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More specifically:
The answer is the starting value of
2. |
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"So, how do we find
inverses?"
Consider the following three
solution methods:
|
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Swap ordered
pairs: If your function is defined as
a list of ordered pairs, simply swap the x and y
values. Remember, the inverse relation will be a function
only if the original function is one-to-one. Examples:
| a. |
Given
function f, find the inverse relation. Is the inverse
relation also a function?
Answer:
Function f
is a one-to-one function since the x and y values
are used only once. Since function f is a
one-to-one function, the inverse relation is also a
function.
Therefore, the inverse function is:
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| b. |
Determine
the inverse of this function. Is the inverse also a
function?
| x |
1 |
-2 |
-1 |
0 |
2 |
3 |
4 |
-3 |
| f
(x) |
2 |
0 |
3 |
-1 |
1 |
-2 |
5 |
1 |
Answer: Swap the x and y
variables to create the inverse relation. The inverse
relation will be the set of ordered pairs:
{(2,1), (0,-2),
(3,-1), (-1,0), (1,2), (-2,3), (5,4),(1,-3)}
Since
function f was not a one-to-one function (the y
value of 1 was used twice), the inverse relation will NOT be
a function (because the x value of 1 now gets mapped to two
separate y values which is not possible for functions).
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Solve
algebraically: Solving for an inverse
relation algebraically is a three step process:
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1. Set the function = y
2.
Swap the x and y variables
3. Solve for
y |
Examples:
| a. |
Find the inverse
of the function 
Answer:
 |
Remember:
Set =
y.
Swap the variables.
Solve for y.
Use the inverse function notation since f (x)
is a one-to-one function. |
|
| b. |
Find the inverse of the
function 
(given that x is not equal to 0).
Answer:
 |
Remember:
Set = y.
Swap the variables.
Eliminate the fraction by
multiplying each side by y.
Get the y's on
one side of the equal sign by subtracting y from each
side.
Isolate the y by factoring out the
y.
Solve for y.
Use the inverse function notation since f (x)
is a one-to-one function.
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Graph:
The graph of an inverse relation is the reflection of the
original graph over the identity line,
y = x. It may
be necessary to restrict the domain on certain functions to
guarantee that the inverse relation is also a function. (Read
more about graphing inverses.)
Example:
| Consider the
straight line, y =
2x + 3, as the original function.
It is drawn in blue.
If reflected over the identity line, y = x,
the original function becomes the red
dotted graph. The new red
graph is also a straight line and passes the vertical line test
for functions. The inverse relation of y = 2x +
3 is also a function.
Not all graphs produce an inverse relation which is
also a function. |
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Use the TI-83+/84+
graphing
calculator
to investigate
inverses.
Click
here. | |
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