Question

What is the domain for the function `f(x)=1/(sqrtx-2)`?

Answer 1

Domain: `[0,4)uu(4,+oo)`
Range:: `(-oo, -0.5]uu(0, +oo)`

Explanation:

`f(x) = 1/(sqrtx-2)`

Considerations for the domain of `f(x)`

`sqrtx` is defined `in RR forall x>=0 ->` Domain of `f(x)>=0`

`f(x)` is undefined at `sqrtx=2 -> x!=4`

Combining these results:
the domain of `f(x) = [0,4)uu(4,+oo)`

Considerations for the range of `f(x)`

`f(0) = -0.5`
Since `x>=0 -> -0.5` is a local maximum of `f(x)`

`lim_(x->4^-) f(x) = -oo`

`lim_(x->4^+) f(x) = +oo`

`lim_(x->+oo) f(x) =0`

Combining these results:
the range of `f(x)=(-oo, -0.5]uu(0, +oo)`

These results can be observed by the graph of `f(x)` below.

graph{1/(sqrtx-2) [-14.24, 14.24, -7.12, 7.12]}