Without finding the inverse of the function determine the range of the inverse of f(x)=sqrt x-4

bear in mind that for any expression, the domain of the expression is the range of its inverse, and the range of the expression is the domain of its inverse.

so for f(x) = √( x - 4 ), let us find its domain, because we want the range of its inverse and they're both the same exact one.

so the domain is the values x can safely takely, in this case, the radicand can't be negative, otherwise we end up with an imaginary value, so whatever values x is, it can't be a value that makes x - 4 less than 0.

a simple way to go about it is, let's simpl set the radicand to 0.

x - 4 = 0, meaning x = 4

if x = 4, then the radicand becomes 0, however if x goes smaller than 4, like say 3, then we have (3) - 4 = -1, and the radicand becomes √(-1), which is not good.

so the domain of f(x) = √(x -4) is all values for x equal to 4 or greater, but not below 4. Incidentally, that is the range of its inverse.