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. Not a function

B. It is a function

C. Not a function

Step-by-step explanation:

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. .

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