![]() Likewise, to indicate that the point where $x=3$ should be excluded from the linear piece (given the strict inequality), we place an open circle at $(3,1)$. Note, that we use a filled-in point at $(3,4)$ to suggest that the output of the function at $x=3$ is determined by the semi-circular piece - since the condition on this piece is true when $x=3$. Then, we discard those points that don't match the conditions provided (i.e., the dashed parts of the left graph below), to arrive at the graph of the piecewise-defined function on the right. We find its graph by first graphing $y = \sqrt$ (a semi-circle with center at the origin and radius 5) and $y=2x-5$ (a line of slope 2 with $y$-intercept at $(0,-5)$), as shown at left below. Graphing a piecewise function can be accomplished by simply graphing the functions found in the respective "pieces", limiting the points drawn for each piece to the $x$-values that satisfy the appropriate condition. Thus, we summarize how to calculate the output with the following "piecewise definition": Multiplying a value by negative one has the same effect - so we can say that for this second piece $|x| = -x$. ![]() ![]() However, when $x \lt 0$, the absolute value changes the sign of its input. When $x \ge 0$, the absolute value function doesn't really do anything - it returns its input unchanged. When one creates a new function from existing functions in a "piecewise-defined" way, one breaks apart some domain into two or more disjoint pieces, using different functions to calculate the output for each $x$-value, where the function used is based upon the piece into which that particular $x$-value falls.Ī simple example of a piecewise defined function is the absolute value function, $|x|$. We can create new functions from existing ones in several ways.
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