Do you understand what "defines a function" means? In order that "f" be a function it only be necessary that a given value of "a" produce a single value of f(a). does NOT "define a function" by y= f(x) because if x= 3, y can be either 4 or -4: and .
But if you are given a function g(x) and then say that f(0)= a and then f(n+1)= g(f(n)), we see:
1) f(0)= a is a given specific number. If we claim that, also, f(0)= b, we can immediately assert, from the definition, that b= a.
2) Suppose that, for some integer k, f(k) is a specific, given number, say f(k)= m. Then f(k+1)= g(f(k))= g(m). Since g is a function, g(m) is a specific number and, therefore, so is f(k+1).
By induction those two statement show that f(n) is a specific number for any positive integer, n.
For example, suppose we define f(0)= 3 and . Then f(1)= g(f(0))= g(3)= 9- 2= 7, f(2)= g(f(1))= g(7)= 49- 2= 47, f(3)= g(47)= 2207, etc. What ever n is, we keep repeating that until we reach f(n). At each step we are applying g to a specific number and, since g is a function, get a specific number as a result. Eventually we get to the specific number that is f(n).