## Inverse Functions

If the ordered pairs of a relation R are reversed, then the new set of ordered pairs is called the inverse relation of the original relation.

##### Example 1

If R = {(1,2), (3,8), (5,6)}, find the inverse relation of R. (The inverse relation of R is written R –1).

R –1 = {(2,1), (8,3), (6,5)}

Notice that the domain of R –1 is the range of R, and the range of R –1 is the domain of R. If a relation and its inverse are graphed, they will be symmetrical about the line y = x.

##### Example 2

Graph R and R –1 from Example along with the line y = x on the same set of coordinate axes.

The answer is shown in Figure 1.

If this graph were “folded over” the line y = x, the set of points called R would coincide with the set of points called R –1, making the two sets symmetrical about the line y = x.

• Identity function. The function y = x, or f (x) = x, is called the identity function, since for each replacement of x, the result is identical to x.

• Inverse function. Two functions, f and g, are inverses of each other when the composition f [ g( x)] and g[ f ( x)] are both the identity function. That is, f [ g( x)] = g[ f ( x)] = x.