Question

Given `f(x) = x/(6+x) and g(x) = (6x)/(1-x)`, how do find `f(g(x)) and g(f(x))`?

Answer 1

`f(g(x))=(6x)/(1-7x)` and `g(f(x))=x`

Explanation:

For `f(g(x))`, we just substitute `x` with `g(x)` in `f(x)` and similarly for `g(f(x))`, we just substitute `x` with `f(x)` in `g(x)`.

As `f(x)=x/(6+x)` and `g(x)=(6x)/(1-x)`

`f(g(x))=((6x)/(1-x))/(1-(6x)/(1-x))`

= `((6x)/(1-x))/((1-x-6x)/(1-x))`

= `(6x)/(1-x)xx(1-x)/(1-x-6x)`

= `(6x)/(1-7x)`

and `g(f(x))=(6xx(x)/(6+x))/(1-x/(6+x))`

= `((6x)/(6+x))/((6+x-x)/(6+x))`

= `((6x)/(6+x))/(6/(6+x))`

= `(6x)/(6+x)xx(6+x)/6`

= `x`