Question

f(x)=1/x-1 and g(x)=2x+4 find (fog)(x) and its domain

Answer 1

\[
f(x) = \frac{1}{x-1} \\
g(x) = 2x + 4 \\
(f \circ g)(x) = f(g(x)) \\
\text{Since }~~f(x) = \frac{1}{x-1} \\
f(g(x)) = \frac{1}{g(x)-1} = \frac{1}{2x+4-1} = \frac{1}{2x+3} \\
\]

Answer 2

In finding the domain of f(g(x)) you have to combine restrictions, if any, on the final function as well as the innermost function in the composite function. The innermost function in (fog)(x) or f(g(x)) is g(x). Here g(x) = 2x + 4 which has no restrictions on its domain. But in the final function, the denominator 2x+3 cannot be zero. So \(x \ne -3/2\).

So the domain is all real values of x except -3/2.