Explanation:
normally a simple expression in x has no natural limitations.
but for fractions we have to pay attention, because we have to avoid results like .../0, as they are undefined.
so, 4x + 1 has no limitations, but x² - 1 must never be 0.
the only "bad cases" are
x² - 1 = 0
x² = 1
x = ±1
therefore, the domain (the definition of the valid values for x) is
x € R, x <> -1 and x <> 1
the range (the definition of valid values for y) is then any value between -infinity and +infinity.
simply because around x=-1 and x=+1 the tendency goes against - and + infinity.