The domain and range of a function `f(x)=(3)/(x)+1` are subsets of A and B respectively, where `A={-(1)/(2),0,(2)/(3),(6)/(7),1}` and `B={-5,0,4(1)/(2),5,5(1)/(2)}`. List the elements of the function as ordered pairs.

Answer

To solve the problem, we need to evaluate the function \( f(x) = \frac{3}{x} + 1 \) at the elements of the domain \( A \) and check if the resulting values are in the range \( B \). The steps are as follows:

### Step 1: Identify the Domain and Range
The domain \( A \) is given as:
\[ A = \left\{ -\frac{1}{2}, 0, \frac{2}{3}, \frac{6}{7}, 1 \right\} \]
The range \( B \) is given as:
\[ B = \left\{ -5, 0, 4\frac{1}{2}, 5, 5\frac{1}{2} \right\} \]

### Step 2: Evaluate \( f(x) \) at Each Element of the Domain
1. **For \( x = -\frac{1}{2} \)**:
\[ f\left(-\frac{1}{2}\right) = \frac{3}{-\frac{1}{2}} + 1 = -6 + 1 = -5 \]
Ordered pair: \( \left(-\frac{1}{2}, -5\right) \)

2. **For \( x = 0 \)**:
\[ f(0) = \frac{3}{0} + 1 \text{ (undefined)} \]
\( 0 \) is not in the domain.

3. **For \( x = \frac{2}{3} \)**:
\[ f\left(\frac{2}{3}\right) = \frac{3}{\frac{2}{3}} + 1 = \frac{3 \cdot 3}{2} + 1 = \frac{9}{2} + 1 = \frac{9}{2} + \frac{2}{2} = \frac{11}{2} \]
This can be written as \( 5\frac{1}{2} \).
Ordered pair: \( \left(\frac{2}{3}, 5\frac{1}{2}\right) \)

4. **For \( x = \frac{6}{7} \)**:
\[ f\left(\frac{6}{7}\right) = \frac{3}{\frac{6}{7}} + 1 = \frac{3 \cdot 7}{6} + 1 = \frac{21}{6} + 1 = \frac{21}{6} + \frac{6}{6} = \frac{27}{6} = \frac{9}{2} \]
This can be written as \( 4\frac{1}{2} \).
Ordered pair: \( \left(\frac{6}{7}, 4\frac{1}{2}\right) \)

5. **For \( x = 1 \)**:
\[ f(1) = \frac{3}{1} + 1 = 3 + 1 = 4 \]
\( 4 \) is not in the range \( B \).

### Step 3: Compile the Ordered Pairs
The valid ordered pairs from the evaluations are:
1. \( \left(-\frac{1}{2}, -5\right) \)
2. \( \left(\frac{2}{3}, 5\frac{1}{2}\right) \)
3. \( \left(\frac{6}{7}, 4\frac{1}{2}\right) \)

### Final Answer
The elements of the function as ordered pairs are:
\[ \left\{ \left(-\frac{1}{2}, -5\right), \left(\frac{2}{3}, 5\frac{1}{2}\right), \left(\frac{6}{7}, 4\frac{1}{2}\right) \right\} \]