Inverse Functions

For a function f(x), the inverse of that function is a function g(y) such that
g(f(x)) = x
Often g(y) is called f-1(y), read "the inverse of f of x."

To find the inverse of a function, y = f(x), just solve for x as a function of y. Some examples of inverse functions:

  1. y = f(x) = x2

  2. y = f(x) = 2x + 4

  3. y = f(x) = sin(cos x)

    arcsin(arccos y) = x = f-1(y)

I've been writing the inverse functions as functions of y, because that way I can say y = f(x) and therefore x = f-1(y). But the important thing is the function, not the variable. It is also correct to say, for example, that g(x) = x - 2 is the inverse function of f(x) = x + 2.

What's the of the inverse of the inverse a function? Or, to ask it another way, for f(x) = y, what's (f-1(y))-1? Well, it would be the function, g(x), such that
g(f-1(y)) = y
But, f-1(y) = x, so we need a function g(x) such that
g(x) = y
This is our definition of f(x), therefore it must be that g(x) = f(x). This agrees with our convention of using f-1(y), since when applied to a variable, (x-1)-1 = x.

I keep saying "inverse function," which is not always accurate. Many functions have inverses that are not functions, or a function may have more than one inverse. For example, the inverse of f(x) = sin x is f-1(x) = arcsin x, which is not a function, because it for a given value of x, there is more than one (in fact an infinite number) of possible values of arcsin x. For x = 0, y can be 0, , ....

Or, for f(x) = x2, both and are valid inverses.

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jjnichol@mit.edu
Last modified 23 June 1997