To understand the concept of inverse function first we must define what is a function. A function is a relationship between two non-empty set of elements called domain (denoted by x variable) and range (denoted by f(x)). A function also requires that for every value of the domain, there exist one and only one element of the range and for every value of the range, there exist at least one value of domain. An example of a function would be f(x)=x2, and the graph is shown in figure 1. Figure 1. Graph of a function f(x)=x2 and x=3
A vertical line test x=k (k is constant), where for this case x=3 (shown in red), is used to test a function. This vertical line verifies that for every value of x (domain) there exist one an only one value of the y (f(x) or range). So the vertical line x=3, intersects the graph of f(x)=x2 only at one point, therefore f(x)=x2 is a function. Test for Existence The inverse of a function exist if the function is one-to-one. A function is said to be one-to-one if for every value in x, sometimes referred to as the domain, there is exist one and only one value of y (range) (Mahler & Wesner, 1994).
So aside from the first requirement for a function which can be tested using vertical line test, another requirement is added for a function to have an inverse. It must therefore be one-to-one, and one famous test for a function to be one-to-one is to use the horizontal line test. A horizontal line is generated by plotting a y=k (k is constant). For a function to be one-to-one, the horizontal line should intersect the function at only one point. So let us go back to the previous plotted equation in Figure 1. We draw a horizontal line y=10 (shown in green), and we can see that the horizontal line intersects the graph at two points.
So f(x)=x2 is not a one-to-one function therefore it has no inverse. An example of a one-to-one function is f(x)=x3 and the graph of the this function is shown in figure 2. Figure 2. Graph of f(x)=x3 Both vertical line x=0 and horizontal line y=1. 6 intersects the graph of f(x)=x3 at only one point, therefore is a one-to-one function. It is possible to obtain the inverse of this function. Obtaining Inverse of a Function To obtain the inverse of a function, we must replace f(x) with y and solve x in terms of y. So that for f(x)=(x/4)3, we solve for f-1(x) as follows
Another interesting characteristic of a function and its inverse is that the inverse of a function is its mirror image with respect to the y=x line. So that any point on the function f(x) has a corresponding reflection on the f-1(x) function. This can be shown in the graphs shown below. Figure 3. Graph of f(x)=(x/4)3 and f-1(x)= 4×1/3 Please note that the gray line is y=x, the blue curve is f(x), and the red curve if f-1(x).
References Wesner, T. H. & Mahler, P. H. (1994). ”Functions”. In College Algebra & Trigonometry with Applications. (pp. 136-137). IOWA : Wm. C. Brown Communications, Inc.