The next objective is to explore the relationships between graphs of *f *and . For this purpose it will be desirable to use *x *as the independent variable for both functions, which means we are comparing the graphs of* y = f*(*x*) and *y* = (*x*).

If (*a, b*) is a point on the graph *y = f*(*x*), then *b = f*(*The*). This is equivalent to the statement that *The* = (*B*) which means that (*b, a*) is a point on the graph of* y* = (*x*).

In summary, inverting the coordinates of a point on the *f *produces a point on the . Similarly invert the coordinates of a point on the graph of produces a point on the* f. *However, the geometric effect of inverting the coordinates of a point is to reflect that point on the line. *y* = *x *(figure 1), and then the graphs of* y = f*(*x*) and *y =* (*x*) are each other relative to this straight line (Figure 2). In short, we have the following result.

If f have a inversethen the graphs of y = f (x) and y = (x) are reflections of each other in relation to the straight y = x; that is, each is the mirror image of the other with respect to that straight line. |

* *

*Increasing or decreasing functions have inverse*

*If the graph of the function f always increasing or always decreasing over the domain of fthen this graph can be cut at most once by any horizontal line and hence the function f must have an inverse.*

*One way to tell if the graph of a function is increasing or decreasing in a range is by examining the inclinations of its tangent lines. The graph of f should be increasing at any interval where f '(x)> 0 (since tangent lines have a positive slope) and should be decreasing at any interval where f '(x) <0 (since tangent lines have negative inclination). These observations suggest the following theorem.*

If the domain of is a range in which f or in which f ' (x)>0 then the function f '(x)<0 there is a f inverse. |

**Example**

The graph of* f*(*x*) = is always growing in , once

for all *x.* However, there is no easy way to solve the equation. *y =* for *x* in terms of *y*; even knowing that* f* There is an inverse, we cannot produce a formula for it.

**NOTE.** What is important to understand here is that our inability to find a formula for the inverse does not negate its existence; indeed, it is necessary to develop ways of finding properties of functions, which have no explicit formula to work with.