Question

How do you find the domain and range algebraically of `f(x)=1/sqrt(x-4)`?

Answer 1

Your function has a square root but also this square root is in the denominator of a fraction.

So, what you wan is to avoid values of `x` that:

1) make the argument of the square root NEGATIVE (you cannot find a real number as solution of a negative square root);
2) make the denominator equal to ZERO (you cannot evaluate a division by zero).

To express the above points mathematically you write that `x-4` MUST be `>0`.

This means that you can accept only values that satisfy:
`x-4>0`
`x>4`

`4` is not acceptable because if you use it you get `sqrt(4-4)` that can be calculated and gives `0` but it would be in the denominator and this is not acceptable.
When you get near `4` your function gets very big tending to `+oo` while going towards `x` values very big the function tends to zero (you can try by yourself substituting values in your function and calculating it).
So your domain will be all the positive `x` bigger than `4` up to `+oo`.
With a range for `y=f(x)` of `+oo` to `0`.
Or graphically:

graph{1/sqrt(x-4) [-10, 10, -5, 5]}