Answer
(a) $f(g(x)) = 3x^2+3x+5$; Domain $(-\infty, \infty)$
(b) $g(f(x)) = 3(3x^2+11x+10)$; Domain $(-\infty, \infty)$
(c) $f(f(x)) = 9x+20$; Domain $(-\infty, \infty)$
(d) $g(g(x)) = x(x^3+2x^2+2x+1)$; Domain $(-\infty, \infty)$
Work Step by Step
$f(x) = 3x+5$
$g(x)= x^2+x$
(a) $f(g(x)) = 3(x^2+x)+5 = 3x^2+3x+5$
Since we have no restrictions, the domain is R, $(-\infty, \infty)$.
(b) $g(f(x)) = (3x+5)^2+3x+5 = 9x^2+30x+25+3x+5 = 9x^2+33x+30 = 3(3x^2+11x+10)$
Since we have no restrictions, the domain is R, $(-\infty, \infty)$.
(c) $f(f(x)) = 3(3x+5)+5 = 9x+15+5 = 9x+20$
Since we have no restrictions, the domain is R, $(-\infty, \infty)$.
(d) $g(g(x)) = (x^2+x)^2+x^2+x = x^4+2x^3+x^2+x^2+x = x^4+2x^3+2x^2+x =
x(x^3+2x^2+2x+1)$
Since we have no restrictions, the domain is R, $(-\infty, \infty)$.
