Question

Can someone explain this? I don't understand a thing. Chapter on functions.

Answer 1

This passage is about the domain of a function. The domain of a function is the set of all points that you can give to the function as an input.

Points not belonging to the domain are not good inputs for the function. In this example, 3 is not a good input, because if you try to evaluate the function at $x=3$, i.e. if you plug 3 in place of x, you get

$f(3) = \dfrac{5}{3-3} = \dfrac{5}{0}$

which is undefined, because you can't divide by 0.

So, when defining this function, you have to specify $x \neq 3$, so that the function will be defined for every other input.

Answer 2

We are given a function of x which is  5 / (x - 3) and it says that  this is only valid  if x is not equal to 3. ( x ≠ 3  is part of the definition of the function).

It can't be 3 because  5 / (3 - 3) = 5/0  and the result of dividing 5 ( in fact any number) by 0 is undefined ( it has no value).

The last sentence  just says that  if part of the definition of the function  is that x is not equal to k then k must be 3.