Part A.
The composition of f ang g is given by
[tex](f\circ g)(x))=f(g(x))=\frac{(3x+7)-7}{3}[/tex]
where we have inserted 3x-7 in the place of x in function f. Then, we have
[tex](f\circ g)(x))=f(g(x))=\frac{3x+7-7}{3}=\frac{3x}{3}=x[/tex]
Therefore, the answer is
[tex](f\circ g)(x))=x[/tex]
Part B
Similarly to the previous case, we have
[tex](g\circ f)(x))=g(f(x))=3(\frac{x+7}{3})-7[/tex]
which gives
[tex](g\circ f)(x))=g(f(x))=x+7-7=x[/tex]
then, the answer is
[tex](g\circ f)(x))=x[/tex]
Part C.
In the first case, x belongs to the domain of g and g(x) belongs to the domain of f. Then, the domain of the composition (fog)(x) is all real numbers.
Similarly, in the second case, x belongs to the domain of f and f(x) belongs to the domain of g. Then, the domain of the composition (gof)(x) is all real numbers. Then, the domains are the same (all real numbers).
