# Pointwise product of functions

Template:New function from old

## Contents

## Definition

### For two functions

Suppose and are functions. The **pointwise product** (often simply called the **product**) of the functions, denoted , or sometimes simply as , is defined as the function:

In other words, every element is sent to the *product* (in the sense of multiplication) of the values of and at that element.

The domain of the pointwise product of and is defined as the intersection of the domain of and the domain of . This is necessary, because, for a pointwise product to make sense, *both* functions must be defined at the point.

### For multiple functions

Suppose are functions. The **pointwise product** of these, denoted or , is defined as the function:

In other words, every element is sent to the *product* (in the sense of multiplication) of the values of at that element.

The domain of the pointwise product of a bunch of functions is defined as the intersection of the domains of all the functions.

## Relation with various operations

### For two functions

Below we discuss how to perform various operations on the pointwise product of and , given knowledge of how to perform the operations on and individually.

Operation | Verbal description | How it's done |
---|---|---|

Graph | We are given the graphs of f and g (without necessarily having algebraic, numerical, or verbal descriptions of the functions) and we need a geometric method to sketch the graph of | too tricky? |

Obtain explicit expression for | We are given algebraic expressions for and and need an explicit algebraic expression for . | Part of a general procedure: finding pointwise combinations of functions by plugging in expressions. See also the piecewise case: finding pointwise combinations of piecewise functions by plugging in expressions. |

Find limit of at a point | We know how to find limits of at points. | limit of products is product of limits |

Differentiate . | We know how to differentiate and individually, we need to differentiate . | product rule for differentiation: . |

Integrate . | We want to integrate in terms of integration of simpler functions. | We can try integration by u-substitution or integration by parts. |