Answer
(a) $$f(g(x)) = sin(x^2+1)$$ $$(-\infty, \infty)$$
(b) $$g(f(x)) = sin^2x+1$$ $$(-\infty, \infty)$$
(c) $$f(f(x)) = sin(sinx)$$ $$(-\infty, \infty)$$
(d) $$g(g(x)) =x^4+2x^2+2$$ $$(-\infty, \infty)$$
Work Step by Step
$f(x)=sinx$
$g(x)=x^2+1$
(a) $f(g(x)) = sin(x^2+1)$
We have no restrictions, so the domain is: $(-\infty, \infty)$
(b) $g(f(x)) = sin^2x+1$
Same here, no restrictions. The domain is: $(-\infty, \infty)$
(c) $f(f(x)) = sin(sinx)$
No restrictions. The domain is: $(-\infty, \infty)$
(d) $g(g(x)) = (x^2+1)^2+1 = x^4+2x^2+1+1 = x^4+2x^2+2$
No restrictions. The domain is: $(-\infty, \infty)$
