Question

Find f(x) and g(x) so that the function can be described as y = f(g(x)). y = 4/x^2+9

Answer 1

I'm assuming all of (x^2+9) is in the denominator. If that assumption is correct, then,

One possible answer is $f(x) = \frac{4}{x}, \ \ g(x) = x^2+9$

Another possible answer is $f(x) = \frac{4}{x+9}, \ \ g(x) = x^2$

There are many ways to do this. The idea is that when we have f( g(x) ), we basically replace every x in f(x) with g(x)

So in the first example above, we would have

$f(x) = \frac{4}{x}\\\\f( g(x) ) = \frac{4}{g(x)}\\\\f( g(x) ) = \frac{4}{x^2+9}$

In that third step, g(x) was replaced with x^2+9 since g(x) = x^2+9.

Similar steps will happen with the second example as well (when g(x) = x^2)