Question

Given the functions f(x) = 2x + 1 and g(x) = x - 3, determine an equation for (f ∘ g)(x) and (g ∘ f)(x).
Determine f(g(5)) and g(f(5)).
Determine all values of x for which f(g(x)) = g(f(x)

Answer 1

(f ∘ g)(x) = f(g) = 2g+1 = 2(x-3)+1 = 2x-5
f(g(5)) = 2*5-5 = 5
or,
f(g(5)) = f(5-3) = f(2) = 2*2+1 = 5

do the same steps for (g∘f)(x)

then, when you have (g∘f) as a function of x, just equate 2x+1 = (g∘f)(x)
Since they are both linear, there will be at most one solution, unless they are the same line.