Question

Find the quotient function f/g for f(x)=sqr(x+1) and g(x)= sqr( x-1).

My Answer:
sqr(x+1)/sqr( x-1)
sqr(x^2-1)/ (x-1)

However, I also have to state the restrictions to the domain and range, which I do not know how to do. Could someone please help me? Thanks.

Answer 1

If Dom(f(x))=domain of f(x), and
Dom(g(x))=domain of g(x), the domain of f(x)/g(x) is the intersection
Dom(f(x))∩Dom(g(x))
However,by the definition of the quotient function, g(x) ≠ 0, therefore, we have to remove the members where g(x)=0 to get the final version:
Dom(f(x)/g(x))
= Dom(f(x))∩Dom(g(x)-{x:g(x)=0}

Note that:
Dom(√(x+1))
= [-1,∞]
Dom(√(x-1))
= [1,∞]
So
Dom(f(x)∩g(x))=[1,∞]
g(x)=0 when x=1, this has to be removed.
Therefore
Dom(f(x)/g(x))
=(1,∞)