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`.