Question

What is the domain of the function?

Answer 1

for a radical with an even root, like 2 in this case, if the radicand turns to a negative value, the result is just an imaginary value, which is another way to say, there's no such a root, no solution per se.

for that, let's first check when the radicand turns to 0 first.

$\bf -8x+8=0\implies -8x=-8\implies x=\cfrac{-8}{-8}\implies \boxed{x=1}$

so, that happens when x = 1, -8(1) +8, is just -8+8 or 0, ok.

so.. if "x" goes a bit higher, like say 2, -8(2) + 8 --> -16 + 8 --> -16

you'd get a negative value, and the radical doesn't have a root for that.

so... the domain, or values "x" can safely take without making the square root expression a negative value, are 1 or below, for example, if we use say -5

-8(-5) + 8 --> 40+8 --> 48  <------ a positive value

thus, (-∞, 1]