A mapping from N to N is defined as follows `F:N rarrN` given by f(n) =`(n+5)^(2),n in N` (N is the set of natural numbers) then

Options

A. f is not one one
B. f is onto only
C. f is both on one and onto
D. f is one one but not onto

Answer

To solve the problem, we need to analyze the mapping defined by the function \( f: \mathbb{N} \to \mathbb{N} \) given by \( f(n) = (n + 5)^2 \). We will determine whether this function is one-to-one (injective) and onto (surjective).

### Step-by-step Solution:

1. **Understanding the Function**:
The function is defined as \( f(n) = (n + 5)^2 \). This means for every natural number \( n \), we will compute \( (n + 5)^2 \).

2. **Finding the Range of the Function**:
Let's calculate the values of \( f(n) \) for the first few natural numbers:
- For \( n = 1 \):
\[ f(1) = (1 + 5)^2 = 6^2 = 36 \]
- For \( n = 2 \):
\[ f(2) = (2 + 5)^2 = 7^2 = 49 \]
- For \( n = 3 \):
\[ f(3) = (3 + 5)^2 = 8^2 = 64 \]
- For \( n = 4 \):
\[ f(4) = (4 + 5)^2 = 9^2 = 81 \]
- For \( n = 5 \):
\[ f(5) = (5 + 5)^2 = 10^2 = 100 \]
The outputs are \( 36, 49, 64, 81, 100, \ldots \).

3. **Checking if the Function is One-to-One (Injective)**:
A function is one-to-one if different inputs produce different outputs.
- Assume \( f(a) = f(b) \) for \( a, b \in \mathbb{N} \).
- This implies \( (a + 5)^2 = (b + 5)^2 \).
- Taking the square root of both sides gives us two cases:
\[ a + 5 = b + 5 \quad \text{or} \quad a + 5 = -(b + 5) \]
- The first case simplifies to \( a = b \), which is valid.
- The second case \( a + 5 = -(b + 5) \) cannot hold since \( a \) and \( b \) are natural numbers (and thus non-negative).
- Therefore, \( f \) is one-to-one.

4. **Checking if the Function is Onto (Surjective)**:
A function is onto if every element in the codomain has a pre-image in the domain.
- The codomain is \( \mathbb{N} \), which includes all natural numbers.
- The outputs we computed are \( 36, 49, 64, 81, 100, \ldots \). These are all perfect squares greater than or equal to \( 36 \).
- Therefore, not every natural number is represented as an output of \( f(n) \) (for example, \( 1, 2, 3, \ldots, 35 \) are not in the range).
- Hence, \( f \) is not onto.

### Conclusion:
- The function \( f(n) = (n + 5)^2 \) is **one-to-one** but **not onto**.