The question provides two functions, $f$ and $g$, in the form of ordered pairs. You are asked to determine the values of compositions of these functions. Let's break it down step by step:

Functions:

$f = \{(6, -2), (8, -1), (10, 0), (12, 1)\}$ $g = \{(-2, 8), (-1, 10), (0, 12), (1, 6)\}$

The compositions and evaluations requested are:

1. $(f \circ g)(1)$
2. $(g \circ f)(6)$
3. $g \circ f \circ g \circ f(1)$

Step 1: $(f \circ g)(1)$

This means we need to find $f(g(1))$.

• From the set $g$, $g(1) = 6$.
• Now we find $f(6)$ from the set $f$, which is $f(6) = -2$.

Thus, $(f \circ g)(1) = -2$.

Step 2: $(g \circ f)(6)$

This means we need to find $g(f(6))$.

• From the set $f$, $f(6) = -2$.
• Now we find $g(-2)$ from the set $g$, which is $g(-2) = 8$.

Thus, $(g \circ f)(6) = 8$.

Step 3: $g \circ f \circ g \circ f(1)$

We need to evaluate this composition step by step:

• First, find $f(1)$, but there is no pair in $f$ for input $1$, meaning this step fails.

Thus, the third composition cannot be evaluated since $f(1)$ is undefined.

Would you like a further breakdown or details on any of the steps? Here are five questions to consider:

1. What is the significance of function composition in mathematical operations?
2. How can function sets with undefined values impact calculations?
3. Can function compositions be applied to non-numeric domains, like strings or sets?
4. What are other methods for expressing functions besides ordered pairs?
5. How do these ordered pair mappings relate to real-world applications?

Tip: When performing function composition, always evaluate from the innermost function outward.