Question

How do you find `f(g(x))` if `f(x) = (x-3) / (5x+1)` and `g(x) = (x-1) / (x^2)`?

Answer 1

`f(g(x))=-(3x^2-x+1)/(x^2+5x-5)`

Explanation:

`f(g(x))`, which can also be written as `(f@g)(x)`, simply means that you take `g(x)`, which equals `(x-1)/x^2`, and replace every instance of `x` with that in `f(x)`.

`f(g(x))=((x-1)/x^2-3)/(5((x-1)/x^2)+1)`

Let's try to clear out fractions by multiplying everything by `x^2`.

`f(g(x))=((x-1)/x^2-3)/(5((x-1)/x^2)+1)(x^2/x^2)=(x-1-3x^2)/(5x-5+x^2)`

Neither the numerator nor the denominator can be factored. The terms can be rearranged to be in descending order, but there's not really anything else you can do from here!