if x is a variable then does it have infinite number of values?
YES, I would say. And here are some examples of why I say this.
--------------------------------------------
Take any logarithmic function:
\(\large\color{black}{ \displaystyle y=\log(x+a)+c }\)
(with any side shift a AND any vertical shift c)
Note: The domain is not impacted by C (and nor is the range since it goes infinitely up and down).
Now, the log is only defined in this case, if \(\large\color{black}{ \displaystyle x+a>0 }\), and thus:
\(\large\color{black}{ \displaystyle x>-a }\), and knowing that no restriction on how large (x+a) can
get your domain is: (-a, +∞)
So if x is over the interval (-a, +∞), then (regardless of the value of a), x takes on infinite number of values.
--------------------------------------------
Obviously, any polynomial is known to be continuous over interval (-∞,+∞).
This way for any polynomial function \(\large f(x)\),
you get that the domain of y=f(x) is
(-∞,+∞)
So, here again x takes on infinite number of values.
--------------------------------------------
Consider any function!! and you will either get that the domain is
`(-∞, +∞)` OR `(s, +∞)` OR `(-∞, s)`
(where s is some number)