Decide whether the given functions are inverses.
\[
f=\{(6,6),(7,6),(8,6)\} ; g=\{(6,6)\}
\]

Are the given functions inverses?
No
Yes

Step 1 ：To determine if two functions are inverses, we need to check if the composition of the two functions results in the identity function. The identity function is a function that always returns the same value that was used as its argument. In other words, for every x in the domain of the identity function, the output is always x.

Step 2 ：The composition of two functions f and g is defined as \((f\circ g)(x) = f(g(x))\) and \((g\circ f)(x) = g(f(x))\).

Step 3 ：Let's check if f and g are inverses:

Step 4 ：Check \((f\circ g)(x) = f(g(x))\):

Step 5 ：Since g only has one point (6,6), g(x) is only defined for x=6. So, we can only check f(g(6)).

Step 6 ：g(6) = 6, so f(g(6)) = f(6). Looking at the function f, we see that f(6) = 6.

Step 7 ：So, \((f\circ g)(x) = x\) for x=6.

Step 8 ：Check \((g\circ f)(x) = g(f(x))\):

Step 9 ：The function f has three points (6,6), (7,6), and (8,6). So, we can check g(f(6)), g(f(7)), and g(f(8)).

Step 10 ：f(6) = 6, f(7) = 6, and f(8) = 6. So, g(f(6)) = g(6) = 6, g(f(7)) = g(6) = 6, and g(f(8)) = g(6) = 6.

Step 11 ：So, \((g\circ f)(x) = x\) for x=6,7,8.

Step 12 ：However, the domain of the identity function is all real numbers, not just {6,7,8}. Therefore, f and g are not inverses of each other.

Step 13 ：\(\boxed{\text{No}}\)