Question

How do you find the domain and range of `g(z) = 1 / sqrt(4-z^2)`?

Answer 1

Here the function is `g(z) = 1/(sqrt(4-z^2)`. But `sqrt(4-z^2` can't be `0` because denominator cannot be `0`. That is because dividing by `0` is not defined.

From this, we can say that `z^2` can't be `4`. That means `z !=sqrt4` `=>` `z!=+-2`.

And the another thing to be noticed is that we can't have a negative number inside a square root. So `z` must be in the range of `2` `&` `-2`.
Now we can say that Domain is all the real numbers between `-2` `&` `2`.

So Domain `= -2` `<` `z` `<` `2, z in RR or (-2,2)`

From this Domain set, we can easily find out Range of the Function.

The Range of the Function is in between `0` `&` `1/sqrt3` including `1/sqrt3`.

So Range `= (0,1/sqrt3]`