Question

How do you find the domain and range of `y = sqrt(x^2 - 1)`?

Answer 1

Range: `y>=0`
Domain: `x>=1` and `x<=-1`

Explanation:

We can first consider the range, rather simply, we must consider all the values that `y` can take on, but as we know `sqrt alpha >=0` ,`alpha in RR`, or in other words, the square root of a value can never be negative for `x in RR`, so hence `y>=0`

Now we can consider the domain, to what we can consider for what values of `x` yields and valid value of `y`, we know the valid values of `sqrt delta` is where `delta >=0`, so in this circumstance we must consider where `x^2-1 >=0` we can sketch to find the values:
graph{x^2-1 [-3.538, 3.572, -1.437, 2.118]}

we can evidently see that where `x^2-1 >= 0` is for `x>=1` and `x<=-1`

So hence the domain is; `x>=1 and x<=-1`