With Maple, explicitly find the inverse of the function

and verify that your answer is the inverse.

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**.