Problem:
With Maple, explicitly find the inverse of the function

and verify that your answer is the inverse.


Visualization:
Using Maple:

First, define the function f:

> f:=x->2/(x^3+1);

Interchange x and y and solve for y:

> solve(x=f(y),y);

There are three solutions but only one of these is real. We now define the function g to be the real solution:

> g:=x->1/x*((-x+2)*x^2)^(1/3);

We plot the two functions f and g and the line y = x. Since - 1 is not in the domain of f, we add the statement discont=true:

> plot({f(x),g(x),x}, x=-3..3, y=-3..3, discont=true);

From this graph, it appears reasonable that g is the inverse of f. The only problem is that the inverse of the part of the blue graph which is pointed to by the purple arrow does not appear where the yellow arrow is pointing. However, since g involves taking the cube root, we have already seen this type of problem before with Maple. To verify that g is the inverse of f, we make the following calculations. First, we check the composition fg:

> (f@g)(x);

> simplify(");

Hence, we have that the composition fg is the identity function. Now we check the composition gf:

> (g@f)(x);

> simplify(");

At first, it appears that the answer is not x; however, a little bit of algebraic manipulation shows that it is, in fact, x. Hence, g is the inverse of f.