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Evaluate piecewise-defined functions with multiple conditions
A piecewise function is defined by different expressions for different parts of its domain. Each "piece" has its own formula that applies when certain conditions are met.
A piecewise function uses different formulas for different parts of its domain. It's written with curly braces showing each expression and the condition when it applies.
Check each condition with your input value. Use the expression for the condition that is true. Only one condition should be satisfied at any given point.
If no condition is satisfied, the function is undefined at that point. Well-defined piecewise functions should cover all possible input values.
< means "less than" (not including the value), while ≤ means "less than or equal to" (including the value). This matters at boundary points between pieces.
Yes, if the pieces connect at their boundaries without jumps or gaps. The function values must match at the transition points for continuity.
Graph each piece separately over its domain. Use open circles for points not included (< or >) and closed circles for points that are included (≤ or ≥).
Yes! Tax brackets, shipping costs, utility rates, parking fees, and many other real-world situations use piecewise functions where different rules apply in different ranges.
The domain is the union of all x-values covered by the conditions. If the conditions cover all real numbers, the domain is (-∞, ∞).
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