Do you mean whether the inverse of a function is a function?

**YES**
Maybe you mean the inverse of #1 is a function and the inverse of #2 is not a function?

**YES**
If all you're checking is to see whether the inverse of a function is a function, then just flip the x and y values to see whether you have a function (you don't have to check for 1-1 adn onto).

To review:

inverse(f) = {<y x> | <x y> in f}

So, inverse(f) is a function if and only if {<y x> | <x y> in f} is a function.

{<y x> | <x y> in f} is a function

if and only if

for all x. y, z, if <y x> and <y z> in {<y x> | <x y> in f} then x = z.

**I DONT UNDERSTAND THIS** **Can you use the conditions and numbers that I have provided to create an example?**

Do you mean f-comp-g?

f-comp-g = {<x y> | exists z such that <x z> in g & <z y> in f}.

So [f-comp-g](x) = f(g(x)). (I.e., compute g(x) = y, then compute f(y).)

**I ALSO DON'T UNDERSTAND THIS please try to help explain given the information I know b/c the notation and the way you are explaining this is above my level of understanding, this is all really new to me, thank you for trying though**