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Intermediate Mathematics - Inverse functions
Inverse functions
The inverse function f(x) of the function f(x) is defined as the function
such that:
f(f(x)) = f(f(x))=x
* The inverse function may be considered as the function that 'undoes'
the original function.
* The range of the inverse function is equal to the domain of the
original function and the domain of the inverse function is equal to the
range of the original function. Thus:
ran f(x) = dom f(x)
dom f(x) = ran f(x)
* To find the rule for the inverse of a function or relation, interchange
x and y in the rule for the function and then rearrange to make y the
subject. This is now the rule, y = f(x) = K , for the inverse.
For example if we wish to find the inverse of f(x) = 5x+1 we write this as
y=5x+1 and then interchange x and y to get
The inverse is
Note that
* The graph of an inverse function is found by reflecting the graph of
the original function in the line y = x.
* The inverse f(x) exists as a function if and only if f is a one-to-one
function.
* It is sometimes necessary to restrict the domain of the original
function so that the inverse exists as a function.
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