With beautiful typesetting (including all the words ‘ if ’ lined up vertically), this author prefers the simplest formatting, with no extra punctuation. The hollow dot at $\,\color\quad -1 \le x < 1\cr There is a rule that applies to the left of $\,x\,$.This is an actual point on the graph of the function it is an (input,output) pair. This produces the solid (filled-in) dot, with coordinates $\,\bigl(x,f(x)\bigr)\,$. A function that is defined by multiple functions on different parts of its domain is called a piecewise defined function. There is a rule that applies at $\,x\,$.In this example, three rules are needed to fully understand what happens at and near $\,x\,$: You may need up to three ‘dots’ to fully understand what's happening at a place where rules are changing. Graphing a Piecewise-Defined Function When a graph is ‘pieced together’ using different rules, as in a piecewise-defined function, A piecewise function is a function in which more than one formula is used to define the output. ![]() ![]() Just use the ‘collapse the graph into the $\,y\,$-axis’ method, as described in this earlier lesson. Once you have a graph, though, the range is easy to obtain: Reading a Piecewise-Defined Function AloudĪloud, it is typically read from left-to-right and top-to-bottom.īy the way, the range of $\,f\,$ is nowhere near as easy to get from the formula. Each of the pieces are specified on a certain domain, where each pair. for inputs in the interval $\,[2,\infty)\,$ the quadratic function $\,x^2\,$ is used for the outputs Piecewise-defined function: A piecewise-defined function is a function that is constructed with 'pieces' of other functions.for inputs in the interval $\,[1,2)\,$ the constant function $\,5\,$ is used for the outputs.for inputs in the interval $\,[-1,1)\,$ the linear function $\,-x 3\,$ is used for the outputs.Finding Function Values for a Piecewise-Defined Function This function $\,f\,$ uses three rules (one for each row) that is, it has three ‘pieces’ (hence the name piecewise-defined function):
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