Answer:
f o g = [tex]\frac{(x+17)^2+(x+2)^2}{(x+2)*(x+17)}[/tex]
(-∞; -17)∪(-17;-2)∪(-2;+∞)
Step-by-step 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)= [tex]\frac{x+17}{x+2}[/tex].
[tex]f(g(x)) = \frac{x+17}{x+2} + \frac{1}{\frac{x+17}{x+2}} =\\ \frac{x+17}{x+2} + \frac{x+2}{x+17} =\\ \frac{(x+17)*(x+17)+(x+2)*(x+2)}{(x+2)*(x+17)} =\\ \frac{(x+17)^2+(x+2)^2}{(x+2)*(x+17)}[/tex]
f o g = [tex]\frac{(x+17)^2+(x+2)^2}{(x+2)*(x+17)}[/tex]
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
[tex]g(x) = \frac{x+17}{x+2} = 0\\x + 17 = 0\\x = -17[/tex]
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;+∞).
