Abstract Algebra Theory and Applications - Computer Science ...

0.2 SETS AND EQUIVALENCE RELATIONS 13words, the inverse function of a function simply “undoes” the function. Amap is said to be invertible if it has an inverse. We usually write f −1 forthe inverse of f.Example 11. The function f(x) = x 3 has inverse f −1 (x) = 3√ x by Example8.Example 12. The natural logarithm and the exponential functions, f(x) =ln x and f −1 (x) = e x , are inverses of each other provided that we are carefulabout choosing domains. Observe thatf(f −1 (x)) = f(e x ) = ln e x = xandwhenever composition makes sense.Example 13. Suppose thatf −1 (f(x)) = f −1 (ln x) = e ln x = xA =( ) 3 1.5 2Then A defines a map from R 2 to R 2 byT A (x, y) = (3x + y, 5x + 2y).We can find an inverse map of T A by simply inverting the matrix A; that is,T −1A= T A−1. In this example,( )A −1 2 −1=;−5 3hence, the inverse map is given byIt is easy to check thatT −1AT −1A(x, y) = (2x − y, −5x + 3y).◦ T A(x, y) = T A ◦ T −1 (x, y) = (x, y).ANot every map has an inverse. If we consider the mapT B (x, y) = (3x, 0)