If f be a function defined as f(x) = p for each `x in R`, where p is a real number, then f is called

Options

A. Constant function
B. Identity function
C. Modulus function
D. Signum function

Answer

To solve the question, we need to analyze the function defined as \( f(x) = p \) for each \( x \in \mathbb{R} \), where \( p \) is a real number.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = p \) indicates that for every input \( x \) from the set of real numbers \( \mathbb{R} \), the output is always the same real number \( p \).

2. **Evaluating the Function for Different Inputs**:
Let's evaluate the function for various values of \( x \):
- For \( x = 1 \), \( f(1) = p \)
- For \( x = 2 \), \( f(2) = p \)
- For \( x = 3 \), \( f(3) = p \)
- Continuing this way, we see that for any real number \( x \), \( f(x) = p \).

3. **Identifying the Nature of the Function**:
Since the output of the function does not change regardless of the input \( x \), we observe that all elements in the domain \( \mathbb{R} \) are mapped to a single element \( p \) in the codomain.

4. **Conclusion**:
A function that assigns the same value \( p \) to every input \( x \) is known as a **constant function**. Therefore, we conclude that the function \( f \) defined as \( f(x) = p \) for each \( x \in \mathbb{R} \) is a constant function.

### Final Answer:
The function \( f \) is called a **constant function**.

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