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

To solve this problem, we need to understand the nature of the functions \( f \) and \( g \) given in the question and how their combination \( f + g \) behaves. 

1. Understanding the Function \( f(a) \):
   • For any natural number \( a \), \( f(a) = \alpha \), where \( \alpha \) is the highest power of any prime \( p \) such that \( p^\alpha \) divides \( a \).
   • This means \( f(a) \) returns the maximum exponent of all prime factors of \( a \).
   • Example: For \( a = 18 = 2^1 \times 3^2 \), \( f(18) = 2 \) because \( 3^2 \) divides 18, and 2 is the highest power among the prime factors.
2. Understanding the Function \( g(a) \):
   • The function \( g(a) = a + 1 \). This means it simply increments the number by 1.
   • Example: If \( a = 4 \), then \( g(4) = 5 \).
3. Combination of Functions \( f + g \):
   • The combined function \( (f + g)(a) = f(a) + g(a) \).
   • Thus, \( (f + g)(a) = \text{maximum power of primes dividing } a + (a + 1) \).
4. Determining if \( f + g \) is One-One and Onto:
   • One-One (Injective): A function is one-one if different inputs produce different outputs.
   • Consider \( a = 8 \) and \( a = 10 \).
     • For \( a = 8 = 2^3 \), \( f(8) = 3 \) and \( g(8) = 9 \), so \( (f + g)(8) = 3 + 9 = 12 \).
     • For \( a = 10 = 2^1 \times 5^1 \), \( f(10) = 1 \) and \( g(10) = 11 \), so \( (f + g)(10) = 1 + 11 = 12 \).
   • Since different inputs (8 and 10) produced the same output (12), \( f + g \) is not one-one.
   • Onto (Surjective): A function is onto if every possible output is the image of at least one input.
   • For \( f + g \) to be onto, every natural number should be obtainable as \( (f + g)(a) \).
   • Due to the specific requirements of \( f(a) \) and \( g(a) \), it's unlikely every number can be expressed as \( f(a) + g(a) \).
   • Therefore, \( f + g \) is neither one-one nor onto.
5. Conclusion:
   • The function \( f + g \) is neither one-one nor onto.

Answer 2

The correct answer is (D):
f, g : N – {1} → N defined as
f(a) = α, where α is the maximum power of those primes p such that p^α divides a.
g(a) = a + 1,
Now, f(2) = 1, g(2) = 3 ⇒ (f + g) (2) = 4
f(3) = 1, g(3) = 4 ⇒ (f + g) (3) = 5
f(4) = 2, g(4) = 5 ⇒ (f + g) (4) = 7
f(5) = 1, g(5) = 6 ⇒ (f + g) (5) = 7
∵ (f + g) (5) = (f + g) (4)
∴ f + g is not one-one
Now, ∵ f_min = 1, g_min = 3
So, there does not exist any x ∈ N – {1} such that
(f + g)(x) = 1, 2, 3
∴ f + g is not onto