I am struggling in understanding the logic behind obtaining the solution of the following problem.
"Consider the two real functions g(x) = x^(-1) and h(y) = y^2+1. What is the composite function? "
I know the answer is (x-1)^(-1/2). But I just don't understand the steps of obtaining this solution. Please help. Thanks
I think that this question is poorly worded. Based on the answer, you want to compose g with the inverse of h. I have several issues with this:
1) It doesn't state in the problem that y is necessarily the dependent variable.
2) The inverse of h is not well defined. In order for the inverse to be well defined the domain of h needs to be restricted, but in this problem there is no restriction on the domain given.
In any case, to "solve" this problem you need to first find "the" inverse of h (it's not unique), and then plug this into g.
x = y^2 + 1 implies x - 1 = y^2 implies y = sqrt(x - 1) and y = -sqrt(x - 1).
Now substitute into g.
Again, for some reason the solution indicates to just substitute the first one into g, but that is not really correct.