# 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 ${f{+}}{g{}}$, ${f{-}}{g{}}$, ${f{{g{}}}}$, and $\frac{{f}}{{g{}}}$ are defined as follows:

## Composition of Functions

Suppose that ${y}={f{{\left({u}\right)}}}={\ln{{\left({u}\right)}}}$ and ${u}={g{{\left({x}\right)}}}={\sin{{\left({x}\right)}}}$. Since ${y}$ is a function of ${u}$ and ${u}$ is afunction of ${x}$ the we obtain that ${y}$ is a function of ${x}$: ${y}={f{{\left({u}\right)}}}={f{{\left({g{{\left({x}\right)}}}\right)}}}={f{{\left({\sin{{\left({x}\right)}}}\right)}}}={\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}$.