A piecewise function is simply two or more functions put together with non-intersecting domains. Once each domain has been determined and the endpoints have been carefully reviewed, then the graph can be completed.
- Determine each sub-domain
Each part of a piecewise function will be valid for certain values of x. Since the domain of the function includes all values that are valid for x, each part of the piecewise function has a sub-domain for which it is valid. For example, the function: f(x) = | x^2 for x ? 0 | 1/x for x < 0 has a domain that includes the entire real number system, but x^2 only has a sub-domain from zero upwards, and 1/x only has a sub-domain of zero downwards but not including zero.
- Know the difference between <, ?, ? and >
In most cases, each part of a piecewise function has different values where the pieces meet. Since this is where each piece ends, it is often called an endpoint. In the above example, x^2 is zero at the endpoint x=0, and 1/x does not exist at that value. Review carefully which piece of the function belongs at that point. When a piece has ? or ?, it belongs at its endpoint and should be represented on the graph by a filled, or closed, circle. A function that does not belong at the endpoint should have a non-filled, or open, circle. In some cases, there may be three (or more) parts to the piecewise function, and there may even be a part that will have a value at a particular point (ex. 3 for x=2). Then each other function that has an endpoint at that value has an open circle.
- Graph the function
Once the first two steps are complete, the function can be graphed with the endpoint labelled correctly for each piece of the function.