Question

find the functions and their domains (enter the domains in interval notation.)f(x)= x + (1/x)g(x)= x + 17 / x+2(a) f o g

Answer 1

f o g =
`((x+17)^2+(x+2)^2)/((x+2)*(x+17))`

(-∞; -17)∪(-17;-2)∪(-2;+∞)

Explanation:

Calculate f o g is equal to do f(g(x)).

To calculate it, we need to replace each x in f(x) with g(x)=
`(x+17)/(x+2)`.

`f(g(x)) = (x+17)/(x+2) + (1)/((x+17)/(x+2)) =\\ (x+17)/(x+2) + (x+2)/(x+17) =\\ ((x+17)*(x+17)+(x+2)*(x+2))/((x+2)*(x+17)) =\\ ((x+17)^2+(x+2)^2)/((x+2)*(x+17))`

f o g =
`((x+17)^2+(x+2)^2)/((x+2)*(x+17))`

The domain of f o g is the set of all real numbers x such that x is in the domain of the function g and g(x) is in the domain of the function f. The domain of g(x) is all the real numbers without -2 because the denominator can't be zero. And the domain of f(x) is all the real numbers without 0 for the same reason.

We need to see when g(x) = 0

`g(x) = (x+17)/(x+2) = 0\\x + 17 = 0\\x = -17`

Therefore, the domain of f o g is all the real numbers without -2 and -17.

Written as an interval is (-∞; -17)∪(-17;-2)∪(-2;+∞).