The following equation:

(f (x)) = x for all x in the field of f |

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 *f*where 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 |

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).*

* Next: Inverse Function Charts
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