For the function h(x)=x+1, where h(x)=(f°g)(x),give the two original functions f(x) and g(x) if the domain of h(x) is [0,∞)

To determine the domain of a composite function, consider the restriction on the domain of the inside function (also referred as the input function) and the restriction on the domain of the final algebraic expression of the composition.

In the given composite function

the final algebraic expression has no restriction on its domain. This indicates that there is only a restriction on the domain of the inside function.

Since

the inside function is g(x).  Take note that if a function has a domain , then it is a radical function and its index is an even number.

So one of the possible form of the inside function is:

The domain of this function is .

In order that  f(g(x)) will be a linear function

the square root in g(x) should be eliminated. So one of the possible form of f(x) is

The domain of this function is .

To verify if these are the individual functions of h(x), take f of g of x.

The resulting algebraic expression is linear. Take note that the domain of a linear function is .

Then, take the intersection of the domain of inside function and final expression.  The two domains and intersect at . So the domain of the composite function h(x) is . 

Therefore, the original functions are   and   .