find a) 𝑓(𝑔(𝑥)) and b) 𝑔(𝑓(𝑥)) and state the domain of each.

Question

If 𝑓(𝑥) = 3/x and 𝑔(𝑥) = 𝑥^2 − 1, find a) 𝑓(𝑔(𝑥)) and b) 𝑔(𝑓(𝑥)) and state the domain of each.

show works please

Answer 1

f(x) = 3/x and g(x) = x² - 1

f(g(x)) = f(x² - 1) = 3 / (x² - 1)

domain of f(g(x)) is { x l x is in dom g and g(x) is in dom f} = {x l x is real and x² - 1 is not zero} = {x l x ≠ ±1}

g(f(x)) = g(3/x) = (9 / x² ) - 1

domain of g(f(x)) = {x l x is in dom f and f(x) is in dom g} = {x l x ≠ 0 and 3 / x is real } = {x l x ≠ 0}