To solve the problem step by step, we will analyze the function \( f = \{(1, 2), (2, -3), (3, -1)\} \) and find the required operations one by one.
### Step 1: Finding \( 2f \)
To find \( 2f \), we multiply the second coordinate of each ordered pair by 2.
- For the first pair \( (1, 2) \):
\[
2 \times 2 = 4 \quad \Rightarrow \quad (1, 4)
\]
- For the second pair \( (2, -3) \):
\[
2 \times (-3) = -6 \quad \Rightarrow \quad (2, -6)
\]
- For the third pair \( (3, -1) \):
\[
2 \times (-1) = -2 \quad \Rightarrow \quad (3, -2)
\]
Thus,
\[
2f = \{(1, 4), (2, -6), (3, -2)\}
\]
### Step 2: Finding \( 2 + f \)
To find \( 2 + f \), we add 2 to the second coordinate of each ordered pair.
- For the first pair \( (1, 2) \):
\[
2 + 2 = 4 \quad \Rightarrow \quad (1, 4)
\]
- For the second pair \( (2, -3) \):
\[
2 + (-3) = -1 \quad \Rightarrow \quad (2, -1)
\]
- For the third pair \( (3, -1) \):
\[
2 + (-1) = 1 \quad \Rightarrow \quad (3, 1)
\]
Thus,
\[
2 + f = \{(1, 4), (2, -1), (3, 1)\}
\]
### Step 3: Finding \( f^2 \)
To find \( f^2 \), we square the second coordinate of each ordered pair.
- For the first pair \( (1, 2) \):
\[
2^2 = 4 \quad \Rightarrow \quad (1, 4)
\]
- For the second pair \( (2, -3) \):
\[
(-3)^2 = 9 \quad \Rightarrow \quad (2, 9)
\]
- For the third pair \( (3, -1) \):
\[
(-1)^2 = 1 \quad \Rightarrow \quad (3, 1)
\]
Thus,
\[
f^2 = \{(1, 4), (2, 9), (3, 1)\}
\]
### Step 4: Finding \( \sqrt{f} \)
To find \( \sqrt{f} \), we take the square root of the second coordinate of each ordered pair. Note that square roots of negative numbers will yield complex numbers.
- For the first pair \( (1, 2) \):
\[
\sqrt{2} \quad \Rightarrow \quad (1, \sqrt{2})
\]
- For the second pair \( (2, -3) \):
\[
\sqrt{-3} = i\sqrt{3} \quad \Rightarrow \quad (2, i\sqrt{3})
\]
- For the third pair \( (3, -1) \):
\[
\sqrt{-1} = i \quad \Rightarrow \quad (3, i)
\]
Thus,
\[
\sqrt{f} = \{(1, \sqrt{2}), (2, i\sqrt{3}), (3, i)\}
\]
### Final Answers
1. \( 2f = \{(1, 4), (2, -6), (3, -2)\} \)
2. \( 2 + f = \{(1, 4), (2, -1), (3, 1)\} \)
3. \( f^2 = \{(1, 4), (2, 9), (3, 1)\} \)
4. \( \sqrt{f} = \{(1, \sqrt{2}), (2, i\sqrt{3}), (3, i)\} \)
