Answer
$a. x$
$b. |x|$
$c. 2$
$d. 26$
$e. 1+x^2$
$f. \sqrt{x-1}$
Work Step by Step
Given $f(x) = 1+x^2$ and $g(x) = \sqrt {x-1}$
a. f o g
$ f (\sqrt {x-1}) = 1 + (\sqrt {x-1})^2 = x$
b. g o f
$g ( 1+ x^2) = \sqrt {1+x^2 - 1} = |x| $
c. (f o g) (2)
$f (g(2)) = f (\sqrt {2-1}) = f(1) = 1 + 1^2 = 2$
d. (f o f) (2)
$f( f(2)) = f(1 + 2^2) = f(5) = 1 + 5^2 = 26$
e. f o g o f
$f(g(f(x))) = f(x) = 1+x^2$
NOTE: g(f(x)) is given from b., so direct substitution is done here.
f. g o f o g
$g(f(g(x))) = g(x) = \sqrt {x-1}$
NOTE: f(g(x)) is given from a., so direct substitution is done here