Piecewise Functions

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What is a piecewise function?

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A piecewise function is a function that is defined by different expressions or formulas for different intervals of its domain.

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How do you evaluate a piecewise function at a specific point?

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To evaluate a piecewise function at a specific point, determine which interval the point falls into and use the corresponding expression to find the function's value.

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What is the domain of a piecewise function?

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The domain of a piecewise function is the set of all possible input values (x-values) for which the function is defined, often specified by the intervals of each piece.

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What is the range of a piecewise function?

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The range of a piecewise function is the set of all possible output values (y-values) that the function can produce, based on the defined pieces.

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How can you graph a piecewise function?

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To graph a piecewise function, plot each piece of the function on its respective interval, ensuring to use open or closed dots based on whether the endpoints are included.

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What does it mean for a piecewise function to be continuous?

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A piecewise function is continuous if there are no breaks, jumps, or holes in the graph at the points where the pieces meet.

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What is the difference between an open interval and a closed interval?

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An open interval does not include its endpoints (e.g., (a, b)), while a closed interval includes its endpoints (e.g., [a, b]).

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How do you find the value of f(x) for x in a specific interval?

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Identify the interval that contains x, then use the corresponding expression for that interval to calculate f(x).

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What is a common mistake when evaluating piecewise functions?

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A common mistake is using the wrong expression for f(x) by not correctly identifying the interval that contains the input value.

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What is the significance of the endpoints in piecewise functions?

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Endpoints determine whether the function includes or excludes certain values, affecting continuity and the overall graph.