For the given function of f: N ->N is defined by f(x)= the sum of all divisors of x, the values of p and (f • f)(p) if p is a prime and f(p) = 5p - 43 are p = 11 and (f • f)(p) = 28.
In the question, we are given that N is the set of natural numbers, and a function of f: N ->N is defined by f(x)= the sum of all divisors of x.
We are asked to find the values of p and (f • f)(p) if p is a prime and f(p) = 5p - 43.
The function of f: N ->N is defined by f(x) = the sum of all divisors of x.
Thus, we know that f(p) = the sum of all divisors of p.
Now, p being prime, it has only two divisors, 1 and p itself.
Thus, we can show that f(p) = 1 + p.
But, we are given that f(p) = 5p - 43.
Thus, equating the two equations, we get:
5p - 43 = 1 + p,
or, 5p - p = 1 + 43,
or, 4p = 44,
or, p = 44/4,
or, p = 11.
Thus, the value of p is 11, and f(p) = 1 + p = 1 + 11 = 12.
We are also asked to find the value of the composite function, (f • f)(p).
We know that (f • f)(p) = f(f(p)).
Also, we know that f(p) = 12.
Thus, (f • f)(p) = f(f(p)) = f(12).
Now, as we know f(x) = the sum of all divisors of x.
Thus, f(12) is the sum of all divisors of 12.
The divisors of 12 are 1, 2, 3, 4, 6, and 12.
Thus, f(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.
Thus, the value of (f • f)(p) = 28.
Thus, for the given function of f: N ->N is defined by f(x)= the sum of all divisors of x, the values of p and (f • f)(p) if p is a prime and f(p) = 5p - 43 are p = 11 and (f • f)(p) = 28.
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