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A piecewise function is usually defined by more than one formula: a fomula for each interval. Example 1: f( x ) = - x if x <= 2 = x if x > 2 What the above says is that if x is smaller than or equal to 2, the formula for the function is f( x ) = -x and if x is greater than 2, the formula is f( x ) = x.
To go from term to term, you keep multiplying by $\,2\,$. CONNECTION BETWEEN GEOMETRIC SEQUENCES AND EXPONENTIAL FUNCTIONS. There is a class of functions, called exponential functions, that have a very special property: equal changes in the input cause the output to be successively multiplied by a constant. Geometric sequences have this same special property: equal changes in the input (e.g., moving from term to term). cause the output to be successively multiplied by a constant (determined by the common ratio). Thus, geometric sequences always graph as points along the graph of an exponential function.