Let `f : {1, 2, 3} rarr {1, 2, 3]` be a function. Then the number of functions `g : {1, 2, 3} rarr {1, 2, 3}`. Such that f(x) = g(x) for atleast one `x in {1, 2, 3}` is

Options

A. 11
B. 19
C. 23
D. 27

Answer

To solve the problem, we need to find the number of functions \( g: \{1, 2, 3\} \to \{1, 2, 3\} \) such that \( f(x) = g(x) \) for at least one \( x \in \{1, 2, 3\} \).

### Step-by-step Solution:

1. **Determine the Total Number of Functions**:
- Each element in the domain \{1, 2, 3\} can map to any of the 3 elements in the codomain \{1, 2, 3\}.
- Therefore, the total number of functions \( f \) from \{1, 2, 3\} to \{1, 2, 3\} is given by:
\[ 3^3 = 27 \]
- The same applies for the function \( g \). So, the total number of functions \( g \) is also:
\[ 3^3 = 27 \]

2. **Calculate the Functions Where \( f(x) \neq g(x) \) for All \( x \)**:
- To find the number of functions \( g \) such that \( f(x) \neq g(x) \) for all \( x \), we can consider the choices for \( g \).
- For each \( x \) in \{1, 2, 3\}, \( g(x) \) must be different from \( f(x) \). Since \( f(x) \) has 1 fixed value, \( g(x) \) has 2 remaining choices.
- Thus, the number of such functions \( g \) is:
\[ 2 \times 2 \times 2 = 2^3 = 8 \]

3. **Calculate the Functions Where \( f(x) = g(x) \) for At Least One \( x \)**:
- The total number of functions \( g \) is 27. The number of functions \( g \) such that \( f(x) \neq g(x) \) for all \( x \) is 8.
- Therefore, the number of functions \( g \) such that \( f(x) = g(x) \) for at least one \( x \) is:
\[ 27 - 8 = 19 \]

### Final Answer:
Thus, the number of functions \( g \) such that \( f(x) = g(x) \) for at least one \( x \) is **19**.