The range and domain of 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 step by step, we will determine the domain and range of the function \( f(x) = \frac{3}{x} + 1 \) and then list the ordered pairs based on the values from the sets \( A \) and \( B \).

### Step 1: Identify the Domain of the Function
The domain of a function consists of all the values of \( x \) for which the function is defined. In this case, the function \( f(x) = \frac{3}{x} + 1 \) is undefined when \( x = 0 \) because division by zero is not allowed.

Given set \( A = \{-\frac{1}{2}, 0, \frac{2}{3}, \frac{6}{7}, 1\} \):
- We can include all values from set \( A \) except \( 0 \).

Thus, the domain of \( f(x) \) is:
\[ \text{Domain} = \{-\frac{1}{2}, \frac{2}{3}, \frac{6}{7}, 1\} \]

### Step 2: Calculate the Range of the Function
Next, we will calculate the range of the function by substituting the values from the domain into \( f(x) \).

1. For \( x = -\frac{1}{2} \):
\[ f\left(-\frac{1}{2}\right) = \frac{3}{-\frac{1}{2}} + 1 = -6 + 1 = -5 \]

2. For \( x = \frac{2}{3} \):
\[ f\left(\frac{2}{3}\right) = \frac{3}{\frac{2}{3}} + 1 = \frac{9}{2} + 1 = \frac{9}{2} + \frac{2}{2} = \frac{11}{2} = 5.5 \]

3. For \( x = \frac{6}{7} \):
\[ f\left(\frac{6}{7}\right) = \frac{3}{\frac{6}{7}} + 1 = \frac{21}{6} + 1 = \frac{21}{6} + \frac{6}{6} = \frac{27}{6} = 4.5 \]

4. For \( x = 1 \):
\[ f(1) = \frac{3}{1} + 1 = 3 + 1 = 4 \]

Thus, the range of \( f(x) \) is:
\[ \text{Range} = \{-5, 5.5, 4.5, 4\} \]

### Step 3: Identify Valid Values in Range and Domain
Now we need to check which of these values fall within the sets \( A \) and \( B \).

Given set \( B = \{-5, 0, 4.5, 5, 5.5\} \):
- The valid outputs from the range that are also in set \( B \) are:
- \( -5 \)
- \( 5.5 \)
- \( 4.5 \)

### Step 4: List the Ordered Pairs
Now we can list the ordered pairs based on the domain and the corresponding range values:

1. From \( x = -\frac{1}{2} \) to \( f\left(-\frac{1}{2}\right) = -5 \):
\[ \left(-\frac{1}{2}, -5\right) \]

2. From \( x = \frac{2}{3} \) to \( f\left(\frac{2}{3}\right) = 5.5 \):
\[ \left(\frac{2}{3}, 5.5\right) \]

3. From \( x = \frac{6}{7} \) to \( f\left(\frac{6}{7}\right) = 4.5 \):
\[ \left(\frac{6}{7}, 4.5\right) \]

4. From \( x = 1 \) to \( f(1) = 4 \) (not in \( B \), so we do not include it):
- Not included.

### Final Ordered Pairs
Thus, the ordered pairs are:
\[ \left(-\frac{1}{2}, -5\right), \left(\frac{2}{3}, 5.5\right), \left(\frac{6}{7}, 4.5\right) \]