# Domain and image of inverse functions

The following equation: (f (x)) = x for all x in the field of f f ( (x)) = x for all x in the field of implies certain relationships between the domains and the images of f and . For example, in the first equation the quantity f (x) is an input of , so points in the images of f are in the domain of ; and in the second equation, the amount (x) is an input of fwhere points in the image of are in the domain of f. All this suggests the following relationships:

 domain of = image of fpicture of = domain of f

Once f and g satisfy two conditions:

• g(f(x)) = x for all x in the field of f
• f(g(y)) = y for all y in the field of g

we conclude that they are inverse. Thus we have the following result.

 If an equation y = f (x) can be resolved to x as a function of y, then f has an inverse and the resulting equation is x = (y)

## A method for finding inverse

Example

Find the inverse of f (x) = Solution. We can find a formula for (y) solving the equation

y = for x as a function of y. The calculations are: of which one has to So far, we have been successful in getting a formula for ; however we are not really complete as there is no guarantee that the associated natural domain is the complete domain for .

To determine if this is what happens, we will examine the image of y = f (x) = . The image consists of all y in the break , so this range is also the domain of (y); soon the inverse of f is given by the formula NOTE. When a formula for is obtained by solving the equation y = f(x) for x as a function of y, the resulting formula has y as the independent variable. If it is preferable to have x as the independent variable for so there are two ways: you can solve y = f(x) for x with a function of y, and then replace y per x in formula Final for , or else you can trade x and y in the equation original and solve the equation x = f(y) for y in terms of x. In this case the final equation will be y = (x).

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