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

1 Answer

Answer:

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]}