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Inverse Functions

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Inverse Functions

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Inverse Functions

Inverse Functions

Inverse Functions

Every one-to-one function f has an inverse function f -1 which essentially reverses the operations performed by f .

More formally, if f is a one-to-one function with domain D and range R , then its inverse f -1 has domain R and range D . f -1 is related to f in the following way: If f (x) = y , then f -1(y) = x . Written another way, f -1(f (x)) = x .


Example: f (x) = 3x - 4 . Find f -1(x) .

The procedure for finding f -1(x) from f (x) involves first solving for x in terms of y .


y   = 3x - 4  
x   =  

Now switch the variables x and y in the equation to generate the inverse:


y   =  
f -1(x)   =  

A function and its inverse are related geometrically in that they are reflections about the line y = x :

Figure %: A function and its inverse are symmetric with respect to the line y = x

Thus, if (a, b) is a point on the graph of f , then (b, a) is a point on the graph of f -1 .

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