gcse edexcel higher functions question

heyy so i'm in year 10 and was doing my maths work from a day i missed in school. there was this question i wasnt really sure how to do so i tried working backwards from the answer and i still couldnt get it so if anyone on here could help that would be great <3 the question is below (sorry formatting isnt great)
f(x) = x + 1 and g(x) = 2x
Let fⁿ (x) mean that you apply the function f n times.
(a) Find fⁿ (x) in terms of x and n.
Answer: x + n
(b) Note that gf²g (x) = 4x + 4. Find all other ways of combining f and g that result in the function 4x + 4.
Answer: g²f, f²gfg, f⁴g²

can you get anywhere with this? Given the definition of f(x) can you work out what f^2 does to x?

since f(x) = x + 1 then surely f²(x) = x + 1 + x + 1 = 2x + 2, meaning the answer to part (a) would be fⁿ(x) = nx + n? but if you go by the actual answer then f²(x) = x + 2. unless "applying the function f n times" means that x remains x and the constant + 1 is the only thing that changes?

For f^2(x), split it up to understand it so
y = f(x)
z = f(y) = f(f(x))
What does z equal?
If you use words to describe the function f(x) = x + 1, then its simply "add one to whatever the input is".

ohhhh okay tysmmmm
mb i didnt realise f²(x) would be a composite function ff(x)
for part (b) would the simplest way to solve it just be trial and improvement or is there a better way to do it?

f^2(x) is slightly ambiguous as it could refer to either f(f(x)) or (f(x))^2 which are obviously different. The context of the question should be enough to say its referring to function composition f(f(x)) and the question states it means apply the function n times. In other cases like sin^2(x) youd interpret it as evaluate sin(x) then square the output so (f(x))^2. Note your interpretation in #2 where
f^2(x) = 2x+2
isnt either of these as
f(f(x)) = x+2
(f(x))^2 = x^2 + 2x + 1
Some simple reasoning/trial and error should be enough. Theres obviously an element of doubling x (twice) and adding one (four times) so just work through/test the different combations. Fairly obviously you have to have a couple of g's, then its just a case of thinking how to get the +4 and the order of the function composition.