Determine if each of the examples below are functions or not. If a given construction is not a function, prove it by showing that a single input can have multiple outputs (not well defined) or that some input doesn't have an output (not total). If it is a function, show it by showing each input has exactly one output.
a) f : R \ {0} -----> R such that f(x) = x^-1
b) f : R -----> R such that f(x) = y iff y <_ x
c) f : Compound Propositions -------> {T,F} such that f(x) = T if x is satisfiable, and f(x) = F otherwise.

A relation from a set X to a set Y is called a function if each element of X is related to exactly one element in Y. That is, given an element x in X, there is only one element in Y that x is related to.

So let check each of the function given.

a. f(x)=x^-1

i.e f(x)= 1/x

Let take a look at the function (1/x) at negative infinity it will have 0 and at positive infinity it will have 0- too.

Therefore it is continuous at both positive and negative infinity

Observe that 1/x→x+∞ as x→0+ and 1/x→x−∞ as x→0−.

Now at x=0 is is infinity and it is not continuous at that point.

Therefore the function is continuous on the real axis except at 0....

If we consider all the points then it is not continuous

b. f(y)= x for y≤x

This is a function because for every value of y there is a unique value of x

1 to 1

2 to 2

3 to 3

And so on

A there is one value for x and y.

Therefore it is a function

c. Given that f(x)= T

And also f(x) = F

This is not a function because a value from the domain has two different value from the range

So it doesn't have a unit value, value x from the domain has two different value F and T from the range.

Then it is not a function. .