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A
function and its inverse 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 of this 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 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.
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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:
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.
Notation:
If f is a given function, then f -1
denotes the inverse of f.
(If the original function is a one-to-one function, the
inverse will also be a function.) |
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If a function is composed with
its inverse,
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 will be a function only if the original
function is one-to-one.
Examples:
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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:
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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:
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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. (Read
more about graphing inverses.)
Example:
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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
of y = 2x + 3 is also a function.
Not all graphs produce an inverse 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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