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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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