Question

What is the domain and range of `y=sqrt(x^2-1)`?

Answer 1

Domain: `(-oo, -1] uu [1, + oo)`
Range: `[0, + oo)`

Explanation:

The domain of the function will be determined by the fact that the expression that's under the radical must be positive for real numbers.

Since `x^2` will always be positive regardless of the sign of `x`, you need to find the values of `x` that will make `x^2` smaller than `1`, since those are the only values that will make the expression negative.

So, you need to have

`x^2 - 1 >=0`

`x^2 >=1`

Take the square root of both sides to get

`|x| >= 1`

This of course means that you have

`x >= 1" "` and `" "x<=-1`

The domain of the function will thus be `(-oo, -1] uu [1, + oo)`.

The range of the function will be determined by the fact that the square root of a real number must always be positive. The smallest value the function can take will happen for `x = -1` and for `x=1`, since those values of `x` will make the radical term equal to zero.

`sqrt((-1)^2 -1) = 0" "` and `" "sqrt((1)^2 -1 ) = 0`

The range of the function will thus be `[0, + oo)`.

graph{sqrt(x^2-1) [-10, 10, -5, 5]}