Question

Let
f,g:N = {1} → N be functions defined by
f(a) = α, where α is the maximum of the powers of those primes p such that p^α divides a, and g(a) = a + 1, for all a ∈ N – {1}. Then, the function f + g is

• A. One-one but not onto
• B. Onto but not one-one
• C. Both one-one and onto
• D. Neither one-one nor onto

Answer 1

The Correct Option is D

The problem involves two functions \( f \) and \( g \) defined on natural numbers \( N \setminus \{1\} \) and asks about the nature of the combined function \( f + g \). Let's analyze these functions step-by-step:

1. For a given number \( a \in N \setminus \{1\} \), the function \( f(a) \) returns \( \alpha \), which is the maximum power of prime \( p \) such that \( p^\alpha \) divides \( a \). This means:
   • If \( a = p_1^{\alpha_1} \times p_2^{\alpha_2} \times \ldots \times p_k^{\alpha_k} \), where \( p_i \) are primes and \( \alpha_i \) are their respective powers in the prime factorization of \( a \), then \( f(a) = \max(\alpha_1, \alpha_2, \ldots, \alpha_k) \).
2. The function \( g(a) \) is straightforward: \( g(a) = a + 1 \).

Now, let's analyze the function \( f + g \):

• The function \( f + g \) should be understood as \( (f + g)(a) = f(a) + g(a) = \alpha + (a + 1) = \alpha + a + 1 \).

To determine the nature of \( f + g \), let's analyze it for both injectivity (one-one) and surjectivity (onto):

• Injectivity (One-One):
  • Consider \( a = 8 = 2^3 \) and \( b = 9 = 3^2 \). For both, the maximum power \(\alpha\): \( f(8) = 3 \) and \( f(9) = 2 \).
  • Thus, \( (f + g)(8) = 3 + 9 = 12 \) and \( (f + g)(9) = 2 + 10 = 12 \).
  • Since \( f + g(8) = f + g(9) \) but \( 8 \neq 9 \), this function is not injective.
• Surjectivity (Onto):
  • To be onto, for every natural number \( n \), there should be an \( a \) such that \( f(a) + g(a) = n \).
  • Let's try to see if \( 2 \) can be expressed as \( f(a) + g(a) \).
    1. If \( f(a) = \alpha \), then \( g(a) = a + 1 \), and that implies \( a + \alpha + 1 = 2 \).
    2. This implies \( a + \alpha = 1 \), which has no natural solution for \( a \neq 1 \). Thus, the function is not surjective.

Hence, the function \( f + g \) is neither one-one nor onto. Therefore, the correct answer is:

• Neither one-one nor onto