1.7  Inverse Functions
Notation
The inverse of the function f is denoted by f
1
(if your browser doesn't support superscripts, that is looks
like f with an exponent of 1) and is pronounced "f inverse". Although the inverse of a function looks like
you're raising the function to the 1 power, it isn't. The inverse of a function does not mean the reciprocal
of a function.
Inverses
A function normally tells you what y is if you know what x is. The inverse of a function will tell you what
x had to be to get that value of y.
A function f
1
is the inverse of f if
•
for every x in the domain of f, f
1
[f(x)] = x, and
•
for every x in the domain of f
1
, f[f
1
(x)] = x
The domain of f is the range of f
1
and the range of f is the domain of f
1
.
Graph of the Inverse Function
The inverse of a function differs from the function in that all the xcoordinates and ycoordinates have
been switched. That is, if (4,6) is a point on the graph of the function, then (6,4) is a point on the graph of
the inverse function.
Points on the identity function (y=x) will remain on the identity function when switched. All other points
will have their coordinates switched and move locations.
The graph of a function and its inverse are mirror images of each other. They are reflected about the
identity function y=x.
Existence of an Inverse Function
A function says that for every x, there is exactly one y. That is, y values can be duplicated but x values
can not be repeated.
If
the function has an inverse that is also a function, then there can only be one y for every x.
A
onetoone
function, is a function in which for every x there is exactly one y and for every y, there is
exactly one x. A onetoone function has an inverse that is also a function.
There are functions which have inverses that are not functions. There are also inverses for relations. For
the most part, we disregard these, and deal only with functions whose inverses are also functions.
If the inverse of a function is also a function, then the inverse relation must pass a vertical line test. Since
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 Spring '10
 BROWN
 Algebra, Inverse Functions, Inverse, Continuous function, Inverse function, Injective function, Inverses

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