Let's analyze the given sets of pairs and extract the unique values in a detailed, step-by-step manner.
We are given two sets of pairs:
1. [tex]\(\{(0,5), (4,2), (6,9), (9,10)\}\)[/tex]
2. [tex]\(\{(5,0), (2,4), (9,6), (10,9)\}\)[/tex]
### Step 1: Extract unique [tex]\(x\)[/tex] values
From the first set of pairs, we extract the [tex]\(x\)[/tex] values:
- For the pair [tex]\((0, 5)\)[/tex], the [tex]\(x\)[/tex] value is [tex]\(0\)[/tex].
- For the pair [tex]\((4, 2)\)[/tex], the [tex]\(x\)[/tex] value is [tex]\(4\)[/tex].
- For the pair [tex]\((6, 9)\)[/tex], the [tex]\(x\)[/tex] value is [tex]\(6\)[/tex].
- For the pair [tex]\((9, 10)\)[/tex], the [tex]\(x\)[/tex] value is [tex]\(9\)[/tex].
Unique [tex]\(x\)[/tex] values from the first set are: [tex]\(\{0, 4, 6, 9\}\)[/tex].
### Step 2: Extract unique [tex]\(f(x)\)[/tex] values
From the second set of pairs, we extract the [tex]\(x\)[/tex] values (these are actually [tex]\(f(x)\)[/tex] values from the first set):
- For the pair [tex]\((5, 0)\)[/tex], the [tex]\(f(x)\)[/tex] value is [tex]\(5\)[/tex].
- For the pair [tex]\((2, 4)\)[/tex], the [tex]\(f(x)\)[/tex] value is [tex]\(2\)[/tex].
- For the pair [tex]\((9, 6)\)[/tex], the [tex]\(f(x)\)[/tex] value is [tex]\(9\)[/tex].
- For the pair [tex]\((10, 9)\)[/tex], the [tex]\(f(x)\)[/tex] value is [tex]\(10\)[/tex].
Unique [tex]\(f(x)\)[/tex] values from the second set are: [tex]\(\{5, 2, 9, 10\}\)[/tex].
### Step 3: Combine and find unique numbers
Now we will combine both unique [tex]\(x\)[/tex] values and unique [tex]\(f(x)\)[/tex] values, and then find all the unique numbers in the combined set.
Combining:
[tex]\( \{0, 4, 6, 9\} \cup \{5, 2, 9, 10\} = \{0, 4, 6, 9, 5, 2, 10\} \)[/tex].
Sorting the combined set, the final unique sequence of numbers is:
[tex]\[\{0, 2, 4, 5, 6, 9, 10\}\][/tex]
### Conclusion
So, the step-by-step results are:
1. Unique [tex]\(x\)[/tex] values: [tex]\[\{0, 2, 4, 5, 6, 9, 10\}\][/tex]
2. Unique [tex]\(f(x)\)[/tex] values: [tex]\[\{0, 2, 4, 5, 6, 9, 10\}\][/tex]
3. Combined unique numbers from both sets: [tex]\[\{0, 2, 4, 5, 6, 9, 10\}\][/tex]
Final results are:
1. [tex]\([0, 2, 4, 5, 6, 9, 10]\)[/tex]
2. [tex]\([0, 2, 4, 5, 6, 9, 10]\)[/tex]
3. [tex]\([0, 2, 4, 5, 6, 9, 10]\)[/tex]
