Let f be a function and let . We often call x the independent variable and y the dependent variable. Now consider specifying a number and working out what x’s are in the domain of the f, for which . There may of course be many such x’s corresponding to a particular y, but functions for which there is only ever one such x have a special name, they are called injections (their old name was one-to-one).

##### Definition

A function is an injection if for all x,y

Or if you prefer

( – for all and – implies.)

Injections are easily recognised from their graphs in that any line parallel to the x axis will meet the graph in at most one point.

Let f be an injection, then we use to denote that single element which is such that .

The function is called the inverse function of the f.

By the definition of we have, .

It is also clear that .

Note that functions that are not injections do not have inverse functions.

ExampleFind the inverse function of the function f given by:

**Solution**: To have a sketch is a great help

So .

Since the rule for f is given in two pieces we find the rule for in two pieces.

.

Whence is given by

Note that f is an injection and so exists then the graphs f and will be reflections of one another in the line y = x.