Question

Let `f(x) = 1/x` and `g(x) = sqrt(x-2)`, how do you find each of the compositions and domain and range?

Answer 1

Please see below.

Explanation:

We cannot have `x=0` in `f(x)`, hence domain is `x!=0`, which can be written in interval notation as `(-oo,0)uu(0,oo)`. As `x` varies in this domain `f(x)` can take all values except `0` andas such range too is `(-oo,0)uu(0,oo)`.

graph{1/x [-10, 10, -5, 5]}

As regards `g(x)=sqrt(x-2)`, we cannot have `x-2<0` i.e. we cannot have `x<2` and hence domain is `[2,oo)`. Given these values of `x`, `g(x)` can take values given by `g(x)>=0` and hence range is `[0,oo)`.

graph{sqrt(x-2) [-4.92, 15.08, -3.36, 6.64]}