Math CentralQuandaries & Queries


Question from Gilligan, a student:

Find functions f and g so that f(g(x)) = H.

(1) H(x) = (1 + x^2)^(3/2)

(2) H(x) = int(x^2 + 1)

I don't know where to start. Can someone answer number 1 with a clear explanation? I should be able to answer number 2 following a reply to question 1. Thanks

Hi Gilligan.

f(g(x)) is a composition function. That means the output of g is used as the input for f.

For example, if g(x) is the number of mosquitos in my yard at a particular time x and f(n) is the rate at which I am getting bitten given the number n of mosquitos in my yard, then f(g(x)) is the rate at which I am getting bitten given the time.

An example with numbers:

H(x) = (1/x)2 + 1

f(g(x)) = H(x)

There are many possible solutions:

  • g(x) = 1/x, f(x) = x2 + 1 or
  • g(x) = (1/x)2, f(x) = g(x) + 1 or even
  • g(x) = x, f(x) = (1/x)2 + 1 (a rather trivial example!)

Basically g(x) is the set of operations you do first and f(x) is the set of operations you perform on the result of g(x) in order to get the final answer H(x).

Hope this helps,
Stephen La Rocque.

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