The functions of a business are the key specialisms that it is divided up into such as marketing, accounts, production, etc. In the past many businesses were divided up on functional lines and this is still the case today in many organisations where it makes sense to create distinct departments in this way.
Many manufacturing organisations are divided up on functional lines. Clearly the most important department in a manufacturing company is often the production department. In individual factories there may be a works or factory manager with overall responsibility for organising and managing the production function.
Then there may be production managers for each of the product lines e.g. biscuits, chocolates etc. However, in addition to the production function there will be other functional specialisms such as the advertising and sales department, the finance and accounts departments, the administration department, the personnel (or human resources) department etc.
Organisations are organised on functional lines when it is felt that this is the best way of communicating and organising the organisation. Specialists are able to focus on their specialism under the supervision of specialist managers.
However, this may not be the only way of organising the organisation. For a number of projects cross-functional teams may be set up. For example, in developing a new product line it may make sense to set up a team containing specialists from marketing, advertising and sales, production, finance and accounts etc.
Once the project is completed these specialists can then go back to concentrating on their own specialist area or they may be pulled into another cross-functional project. When specialists have more than one line manager they may be organised in a matrix structure. For example, when projects are set up a production line worker may both be accountable to the production manager and to the project leader for new product development.
A matrix is a structure with more than one line of command. Individuals who make up the matrix may be accountable to two or more supervisors/managers depending on the complexity of the matrix.