What kind of function has an inverse function? In high school, we once introduced that when a function is a one-to-one mapping , the function and its inverse function are inverse functions of each other . Below we further discuss the sufficient conditions for a function to have an inverse function.

　　In general, even if a function is single-valued , its inverse

It doesn't have to be a single value either. As shown in Figure 1, we can take a point in the value domain of the function and draw a straight line parallel to axis. There are two intersections between this straight line and the curve , and their abscissas are and respectively .

Figure 1：[ Animation ]

　　But if the function is not only single-valued but also monotonic on the interval , then its inverse function is in

The above is single value. In fact, if is a monotonic function on , then for any two different values , ( ) on , there must be

.

　　Therefore, when taking any value on , there cannot be two different values and on , so that

and

Established at the same time. Therefore, the fact that function is single-valued and monotonic on the interval  is a sufficient condition for the existence of its inverse function .

　　For example, the domain of the quadratic function is and the value range is . It is easy to see that the inverse function of is not a single-valued function. But the function increases monotonically in the interval ( see Figure 2 ) , so when is limited to , the inverse function of the function is a single-valued function ( see the figure below ) , that is

.

In the same way, the inverse function of function on the interval is also single-valued, that is

.

Figure 2：[ Animation ]