# Category: Creating New Functions from Old

## Transformation of Functions

By applying certain transformations to the graph of a given function we can obtain new functions. This will give the ability to sketch the graphs of many functions quickly based on the old one. It will also be easier to write equations for given graphs.

## 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:

- `(f+g)(x)=f(x)+g(x)`. Domain is intersection of domains `X_1` and `X_2`: `X_1 nn X_2`.
- `(f-g)(x)=f(x)-g(x)`. Domain is intersection of domains `X_1` and `X_2`: `X_1 nn X_2`.
- `(fg)(x)=f(x)*g(x)`. Domain is intersection of domains `X_1` and `X_2`: `X_1 nn X_2`.
- `(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`.

## Composition of Functions

Suppose that `y=f(u)=ln(u)` and `u=g(x)=sin(x)` . Since `y` is a function of `u` and `u` is afunction of `x` the we obtain that `y` is a function of `x`: `y=f(u)=f(g(x))=f(sin(x))=ln(sin(x))` .

The procedure is called composition because the new function is composed of the two given functions `f` and `g`.