A Step Function That Jumps Once and Drops

This applet demonstrates the behavior and graphical representation of the step function f(x) = (x/|x| + (1-x)/|1-x|) / 2. The function takes the value 1 on the interval [0, 1) and equals 0 everywhere else, producing a distinctive single jump up followed by a drop back down. On the left side of x=0, both x and 1-x are positive when x<0, yielding f(x)=0. At x=0, the expression is specially handled so that f(0)=1. Throughout the open interval (0,1), x is positive and 1-x is positive, giving f(x)=1. At x=1, the term (1-x)/|1-x| becomes undefined in the algebraic form, but the piecewise convention assigns f(1)=0. Beyond x=1, both terms flip sign appropriately and f(x)=0 again. The applet allows students to interactively explore how an algebraic formula involving absolute values and division can encode a piecewise-defined function without explicitly writing out cases. A toggle or alternative definition using GeoGebra's If command is also displayed, showing the equivalent piecewise notation and reinforcing the connection between algebraic and conditional definitions. By manipulating the visualization, learners can observe the discontinuities at x=0 and x=1, verify the function values across different domains, and compare the absolute-value construction with the If-based construction side by side. This exercise supports key learning objectives in understanding piecewise functions, including recognizing jump discontinuities, interpreting absolute value sign behavior, translating between algebraic and conditional representations, and accurately graphing step functions. It is particularly useful for high school or early college students encountering piecewise-defined functions for the first time and seeking a deeper intuitive grasp of how formulas can encode different rules on different intervals.