Piecewise graphs and endpoints

A piecewise function gives different rules on different input intervals. Read the interval before substituting. At a shared boundary, a filled point means that value is included and an open point means it is excluded. Two pieces may approach different heights at the same input; only the rule whose interval includes the input defines the actual function value.

Find the range of each piece separately and combine the sets of outputs. An open input endpoint need not create an open range endpoint: the same output might occur elsewhere. A quadratic piece may attain a minimum inside its interval. Keep the graph within the stated domains and check a value on either side of each boundary.

Worked example

For −2 ≤ x < 0 let f(x) = x², and for 0 ≤ x ≤ 2 let f(x) = x + 1. Describe the endpoints, find f(0) and find the range.

1. The quadratic piece includes (−2,4) and excludes (0,0); its outputs satisfy 0 < y ≤ 4.
2. The line includes (0,1) and (2,3); its outputs satisfy 1 ≤ y ≤ 3.
3. The input 0 belongs to the line piece, so f(0) = 1. Combine the two output sets.

Answer: f(0) = 1; range 0 < f(x) ≤ 4. The graph has a jump at x = 0.

Revise first: Domain and range.