# Combinations of Functions

Let and f and g be functions with domains X_1 and X_2. Then the functions f+g, f-g, fg, and f/g are defined as follows:

1. (f+g)(x)=f(x)+g(x). Domain is intersection of domains X_1 and X_2: X_1 nn X_2.
2. (f-g)(x)=f(x)-g(x). Domain is intersection of domains X_1 and X_2: X_1 nn X_2.
3. (fg)(x)=f(x)*g(x). Domain is intersection of domains X_1 and X_2: X_1 nn X_2.
4. (f/g)(x)=(f(x))/(g(x)). Domain is intersection of domains X_1 and X_2: and such x that g(x)!=0: {x in X_1 nn X_2,g(x)!=0}.

Example. If f(x)=sqrt(x-2) and g(x)=sqrt(9-x^2) find f+g, f-g, fg, and f/g.

Domain of f(x) is x-2>=0 or interval [2,oo). Domain of g(x) is 9-x^2>=0 or interval [-3,3].

So, the intersection of domains is [2,\ oo] nn [-3,\ 3]=[2,\ 3].

Thus,

(f+g)(x)=f(x)+g(x)=sqrt(x-2)+sqrt(9-x^2) for 2<=x<=3.

(f-g)(x)=f(x)-g(x)=sqrt(x-2)-sqrt(9-x^2) for 2<=x<=3.

(fg)(x)=f(x)g(x)=sqrt(x-2)sqrt(9-x^2)=sqrt((x-2)(9-x^2)) for 2<=x<=3.

(f/g)(x)=f(x)/g(x)=sqrt(x-2)/sqrt(9-x^2) for 2<=x<3.

Notice that the domain of f/g is the interval [2, 3) because we must exclude the points where g(x)=0, i.e. x=+-3.