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Applications to Graphing
- Inverse Functions |
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Composition can be used to express the
relationship between
a function and its inverse.
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If f is a given
function and f -1 denotes the inverse of f,
then
and
(where x denotes the identity function).
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"When
you compose a function with its inverse,
you end up with the same value that you used to start! If you use 2
as the x-value in the composition of a function and its inverse, you will
get 2 as the answer!"
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Example:
g(x) = x + 2 and
h(x) = x - 2 are inverse functions because:
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Remember
that an inverse is always a function. |
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Method 1 -
Determine graphically if a
function has an inverse:
Use the
horizontal line test
to determine if a function has an inverse.
If ANY horizontal line intersects your original function in ONLY ONE
location, your function has an inverse (which is a function).
The function y = 3x + 2, shown at the right, HAS
an inverse because it passes the horizontal line test.
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Method 2 -
Determine graphically if a
function has an inverse: If a
function has an inverse, the
reflection of that original
function in the identity line y = x will be a function (it
will pass the vertical line test for functions).
The example at the left shows the original
function, y = x2 , in blue.
The reflection over the identity line y = x is shown in
red. The
red dashed line will not pass the
vertical line test for functions, thus y = x2 does not
have an inverse. |
NOTE:
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 selected section of the
original graph that will pass the horizontal line test for
the existence of an inverse. For
example:
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or
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by restricting the graph in such a manner, you
guarantee the existence of an inverse for a portion of the
graph. (Other restrictions are also possible.) |
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