Functions and domains

Question

I hope I am going in the right direction

f(x)= 2x+1     g(x) =3x-2

 

(f-g)(x)=2x+1-(3x-2)=-x +3, domain=(indefinite sign,3)

 

(f x g)(x)= (2x+1)(3x-2)= 6x² -x-2, domain =?

 

Answer 1

You correctly combined the two functions.

 

In both cases the domain is all real numbers or {-∞<x<∞}, as you can input any x. The two cases which limit the domain are when you divide by zero, or when you have a root of a negative (neither of which apply here).

 

The range is all real numbers for the first as its a line which goes from  -∞ to ∞ in both the x and y directions.

The range is limited in the second one because of the x². If it was just an x² you could easily see the y-values never go below zero. This is complicated by the -x-2 which is added on. Using calculus it can be seen that the slope is zero at x=1/12, resulting in a minimum there. This corresponds to y=-49/24. So the range is (-49/24<y<∞}  ; where () include the number like a ≤, and {} include everything up to that number like a <.