Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

Solution:

If f(x) = g(x) = 1/x then they are inverse of each other. Let us see how? As we have defined above

f(x) = 1/x --- (1)

And also

g(x) = 1/x --- (2)

Therefore substitute x with g(x) in equation (1) we have,

f(g(x)) = 1/1/x = x --- (3)

Now substitute x with f(x) in equation (2) to get

g(f(x)) = 1/1/x = x --- (4)

As per the problem statement the condition that both functions are inverse of each other means the following:

f(g(x)) = x and g(f(x)) = x.

In equations (3) and (4) we have just proved that and hence we conclude that when f(x) = 1/x = g(x) then they are both inverses of each other.

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

Summary:

By choosing f(x) = g(x) =1/x they become inverse of each other and this is confirmed in the solution by verifying that f(g(x)) = g(f(x)) = x.