Let X ={1,2,3,4,5} and
Y={6,7,8,9,10,11,12,13,14,15} if a funtion f : X `rarr` Y defined by f(x)=2x+5 then

Options

A. f(f(1)) is not defined
B. f(f(2)-7)=f(2)
C. f(7)=2
D. f(2)=f(7)

Answer

To solve the problem step by step, we need to evaluate the function \( f \) defined by \( f(x) = 2x + 5 \) for the elements in the set \( X = \{1, 2, 3, 4, 5\} \) and check if the output values are in the set \( Y = \{6, 7, 8, 9, 10, 11, 12, 13, 14, 15\} \).

### Step 1: Evaluate \( f(1) \)
- We substitute \( x = 1 \) into the function:
\[ f(1) = 2(1) + 5 = 2 + 5 = 7 \]
- Check if \( 7 \) is in \( Y \):
- Yes, \( 7 \in Y \).

### Step 2: Evaluate \( f(f(1)) \)
- We need to find \( f(f(1)) = f(7) \).
- However, \( 7 \) is not in the domain \( X \) (since \( X = \{1, 2, 3, 4, 5\} \)).
- Therefore, \( f(f(1)) \) is not defined.

### Step 3: Evaluate \( f(2) \)
- Substitute \( x = 2 \) into the function:
\[ f(2) = 2(2) + 5 = 4 + 5 = 9 \]
- Check if \( 9 \) is in \( Y \):
- Yes, \( 9 \in Y \).

### Step 4: Evaluate \( f(f(2)) \)
- We need to find \( f(f(2)) = f(9) \).
- However, \( 9 \) is not in the domain \( X \).
- Therefore, \( f(f(2)) \) is not defined.

### Step 5: Evaluate \( f(3) \)
- Substitute \( x = 3 \) into the function:
\[ f(3) = 2(3) + 5 = 6 + 5 = 11 \]
- Check if \( 11 \) is in \( Y \):
- Yes, \( 11 \in Y \).

### Step 6: Evaluate \( f(f(3)) \)
- We need to find \( f(f(3)) = f(11) \).
- However, \( 11 \) is not in the domain \( X \).
- Therefore, \( f(f(3)) \) is not defined.

### Step 7: Evaluate \( f(4) \)
- Substitute \( x = 4 \) into the function:
\[ f(4) = 2(4) + 5 = 8 + 5 = 13 \]
- Check if \( 13 \) is in \( Y \):
- Yes, \( 13 \in Y \).

### Step 8: Evaluate \( f(f(4)) \)
- We need to find \( f(f(4)) = f(13) \).
- However, \( 13 \) is not in the domain \( X \).
- Therefore, \( f(f(4)) \) is not defined.

### Step 9: Evaluate \( f(5) \)
- Substitute \( x = 5 \) into the function:
\[ f(5) = 2(5) + 5 = 10 + 5 = 15 \]
- Check if \( 15 \) is in \( Y \):
- Yes, \( 15 \in Y \).

### Step 10: Evaluate \( f(f(5)) \)
- We need to find \( f(f(5)) = f(15) \).
- However, \( 15 \) is not in the domain \( X \).
- Therefore, \( f(f(5)) \) is not defined.

### Conclusion
- The function \( f(f(x)) \) is not defined for any \( x \in X \) because \( f(x) \) always produces values that are not in the domain \( X \).
- Thus, the answer is that \( f(f(1)) \), \( f(f(2)) \), \( f(f(3)) \), \( f(f(4)) \), and \( f(f(5)) \) are all not defined.