Question

What is the domain and range of `f(x) = sqrt((x^2) - 3)`?

Answer 1

Domain: `x<-sqrt3, x>sqrt3`

Range: `f(x)>=0`

Explanation:

I'm going to assume for this question that we are staying within the realm of Real Numbers (and so things like `pi` and `sqrt2` are allowed but `sqrt(-1)` is not).

The Domain of an equation is the list of all allowable `x` values.

Let's look at our equation:

`f(x)=sqrt(x^2-3)`

Ok - we know that square roots can't have negative numbers in them, so what will make our square root term negative?

`x^2-3<0`

`x^2<3`

`x < abssqrt3 => -sqrt3< x< sqrt3`

Ok - so we know that we can't have `-sqrt3< x< sqrt3`. All other `x` terms are ok. We can list the domain in a few different ways. I'll use:

`x<-sqrt3, x>sqrt3`

The Range is the list of resulting values coming from the domain.

We already know that the smallest number the range will be is 0. As `x` gets larger and larger (both in a positive and negative sense), the range will increase. And so we can write:

`f(x)>=0`

We can see this in the graph:

graph{sqrt(x^2-3) [-10,10,-2,7]}