Find fog and gof. Also, find the domain for each of the two compositions (write in interval notation.)

Question

f(x)= 4/x²-1 and g(x)= x+1

Answer 1

for f(x) =4/xsq -1 we have vertical asymptotes at -1 and 1 so domain is ( -infinity,-1) (-1,1), (1-infinity)

for g(x)=x+1 domain is all reals(-infinity,infinity)

Answer 2

f(x) = 1 / (x²-1)

g(x) = x + 1

(fog)(x) = f(g(x)) = f(x+1) = 1 / [(x+1)²-1] = 1 / (x²+2x)

domain(fog) = {x l x ε dom(g) and g(x) ε dom(f)} = {x l x+1 ≠ ±1} = {x l x ≠ 0, -2}

(gof)(x) = g(f(x)) = g(1 / (x²-1)) = 1 / (x²-1) + 1 = x² / (x² - 1)

dom(gof) = {x l x ε domf and f(x) ε domg} = {x l x ≠ ±1}