Question

Let `f(x)= 1-x^3` and `g(x)= 1/x`, how do you find each of the compositions?

Answer 1

Think of the `x` as just a place holder for something to get:

`(f*g)(x) = f(g(x)) = 1-(g(x))^3 = 1-(1/x)^3 = 1-1/x^3`

`(g*f)(x) = g(f(x)) = 1/f(x) = 1/(1-x^3)`

Explanation:

An equation like `f(x) = 1 - x^3` basically describes what you do to any thing to turn it into `f` of that thing.

You can replace `x` with any number or expression and the equation remains true.

So to find `(f*g)` just substitute the expression for `g(x)` into the equation for `f(x)` to get:

`(f*g)(x) = f(g(x)) = 1-(g(x))^3 = 1-(1/x)^3 = 1-1/x^3`

Similarly,

`(g*f)(x) = g(f(x)) = 1/f(x) = 1/(1-x^3)`