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Definition of Inverse Function |
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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. It is also true that the
inverse of a function may not necessarily
form another function.
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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. |
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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:
For all one-to-one functions, the inverse
function is the set of ordered pairs obtained by
interchanging the first and second elements of each pair in the
original function.
Notation:
If f is a given function, then f -1
denotes the inverse of f. |
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"So, how do we find
inverse functions?"
Consider the following:
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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 will be a function only if the original
function is one-to-one.
Examples:
| a. |
Given function f,
find the inverse. Is the inverse also a function?:
Answer:
Function f is a one-to-one function since the x
and y values are used only once. The inverse is
Since function f is a one-to-one function, the inverse is also
a function.
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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 |
Answser: Swap the
x and y variables to create the inverse. Since
function f was not a one-to-one function (the y
value of 1 was used twice), the inverse 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).
| x |
2 |
0 |
3 |
-1 |
1 |
-2 |
5 |
1 |
| f-1(x) |
1 |
-2 |
-1 |
0 |
2 |
3 |
4 |
-3 |
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Solve
algebraically:
Solving
for an inverse algebraically is a three step process:
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:
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Remember:
Set = y.
Swap the variables.
Solve for y. |
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| b. |
Find the inverse of
the function
Answer:
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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.
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Graph:
The graph of an inverse 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 is also a function.
Example:
Consider, as our original function:
y = x2.
This original function is denoted in
blue.If
reflected over the identity line y = x, the original function
becomes the red dashed graph.
Since the
red graph will not pass the
vertical line test for functions, our original function,
y = x2, does not
have an inverse function.
You can see that the inverse exists, but it is NOT a function.
With functions such as y = x2
, it is possible
to restrict the domain to obtain an inverse function
for a portion of the graph. This means that you will be
looking at only a |
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selected section of the
original graph that will pass the horizontal line test for
the existence of an inverse function. For
example, restrict such as:: |
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or
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by restricting the graph in such a manner, you
guarantee the existence of an inverse function for a portion of the
graph.
(Other restrictions are also possible.) |
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