Question

Find the functions and their domains. (Enter the domains in interval notation.)

f(x) = x +(1/x
g(x)= (x+8)/(x+2)
f ∘ g
(f ∘ g)(x) = _____
domain = _____

Answer 1

To find f ∘ g (f ∘ g)(x), substitute the expression for g(x) into f(x) and simplify the resulting expression. The domain of f ∘ g is determined by the domain of g, which is all real numbers except x = -2.

Step-by-step explanation:

To find f ∘ g (f ∘ g)(x), we need to substitute the expression for g(x) into f(x).

First, let's find g(f(x)):

g(f(x)) = g(x + 1/x) = [(x + 1/x) + 8]/[(x + 1/x) + 2]

Now, substitute this expression into f(x):

f(g(x)) = f([(x + 1/x) + 8] / [(x + 1/x) + 2])

Now simplify:

f(g(x)) = ([(x + 1/x) + 8] / [(x + 1/x) + 2]) + 1/([(x + 1/x) + 8] / [(x + 1/x) + 2]))

So, f ∘ g (f ∘ g)(x) = ([(x + 1/x) + 8] / [(x + 1/x) + 2]) + 1/([(x + 1/x) + 8] / [(x + 1/x) + 2]))

The domain of f ∘ g is determined by the domain of g, which is all real numbers except x = -2 (since division by zero is undefined).