Thread: Composite function problem

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

Originally Posted by Real9999

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

Are you working out the composite or inverse composite function ?

There are two composite functions h(g(x)) and g(h(y))

You compose two functions just by substituting one in for the variable in the other.

For example, h(g(x)) = h(x^(-1)) = (x^(-1))^2+1 = x^(-2) + 1 = 1/x^2 + 1 = (1 +x^2)/x^2

Originally Posted by Real9999

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

Would it help you to understand that the first step would be to take h(y) = y^2+1 and substitute x to get h(x) = x^2 + 1?

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.