Inverse Functions
A function
is said to be one-to-one if
whenever
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Here is a one-to-one function
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While the function below is not one-to-one.
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¡@
Horizontal Line test
A function is one-to-one if and only if no horizontal line intersects its graph more than once. The animation below shows that the function
is one-to-one.
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¡@
If we reverse the inputs and outputs of the function F below,
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we get a new function G as follows .
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Notice that for all
in the domain of F , we have
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and for all
in the domain of G , we have
.
Letbe a one-to-one function with domain A and range B . Then the inverse function
has domain B and range A and is defined by
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for any
in B.
This says that ifmaps
to
, then its inverse function
maps
back to
.
In the example above , G is the inverse function of F .
Does function H have an inverse function? Why?
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How to find the inverse function
of a one-to-one function
?
Step 1 Write
.
Step 2 Solve this equation forin terms of
.
Step 3 To expressas a function of
, interchange
and
, the resulting equation is
.
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For example, if we consider the function
. ( Note that is one-to-one.)
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Solving the equation
for
in terms of
,
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we get that the inverse function is
.
The following figure shows the graphs of(red) and
(blue). Note that they are mirror images of each other through the line
(green).
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¡@
Now we consider what does it mean to invert the sine function. First we note that there is a complication with trigonometric function that did not arise with the functions we have just discussed. The sine function clearly fails the horizontal line test.
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¡@The following figure shows two periods of the sine function (blue) and the result (red) of interchanging
and
in the equation
.
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¡@
The problem is that the red curve is obviously not the graph of a function. In order to have an invertible function that takes all the values of the sine function, we restrict the domain of the sine to the interval from
to
.
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This restricted sine and its inverse (called arcsine , in Maple we use the command arcsin(x) ) are shown in the next figure.
(red)
(blue)
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¡@
Here are problems to think about:
If we restrict the domain of sine to [
,
], what does the resulting inverse function look like? What is the relation between this inverse function and
?
By restricting the domain of cosine to the interval from
to
, we invert cosine in the same fashion. The resulting inverse function is called arccosine. Note that the domain of both
and
are [
, 1 ].
What is the relation between
and
?
If
and
, the exponential function is either increasing or decreasing and so it is one-to-one. It therefore has an inverse function which is called the logarithmic function with base
and is denoted by
. Hence
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and
for every
,
for every
.
Laws of Logarithms
1.
2.![]()
3.(where
is any real number)
The logarithm with base
is called the natural logarithm and has special notation
.
Hence, we have
for every
,
for every
.
In particular,
.
The following figure shows the graphs of(red) and
(blue). Note that they are mirror images of each other through the line
(green).
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