HELP! I can't do this problem!?

Question

if f(x)=x^2+1 and g(x) =x+1, then the solution set of
f(g(x))=g(f(x)) is
a) {0,1}
b) {0}
c) all reals
d) {x: x is greater than 0}
e) not equal to zero

please show mw how to do the problem

Answer 1

Only 0 and 1 will satisfy the solution set you're looking for. (Just try any other number, you'll see.) If you make x^2+1 = x+1, then subtract 1 from both sides, you get x^2 = x. Next, factor both sides, and you get x*x = x*1, so either x = 0 or x = 1. (Answer a)

Answer 2

With f(x) = x^2 + 1 and g(x) = x + 1, we have

f(g(x)) = (g(x))^2 + 1 = (x + 1)^2 + 1= (x^2 + 2x + 1) + 1
= x^2 + 2x + 2,

and

g(f(x)) = f(x) + 1 = (x^2 + 1) + 1 = x^2 + 2.

Now suppose they are equal: x^2 + 2x + 2 = x^2 + 2. This reduces to 2x = 0, or x = 0. Thus, the solution set is {0}.