Function: Domain and Range, Methods of Representation
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What is a function in mathematics?
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A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
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What is the domain of a function?
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The domain of a function is the complete set of possible input values (x-values) which produce a valid output from the function.
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What is the range of a function?
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The range of a function is the set of all possible output values (y-values), which result from using the function.
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How is a function typically represented?
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A function can be represented as an equation, a graph, a table of values, or a mapping diagram.
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Explain one way to represent a function graphically.
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Graphically, a function is represented by plotting it on a coordinate plane, typically as a line or curve.
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What is an explicit equation in the context of a function?
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An explicit equation is a mathematical expression that clearly defines the relationship between variables, generally written as y = f(x).
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How can a table be used to represent a function?
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A table represents a function by listing input values (domain) alongside their corresponding output values (range).
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What is a mapping diagram?
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A mapping diagram depicts a function by showing how each element of the domain is related to an element in the range using arrows.
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Why is it important to specify the domain of a function?
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Specifying the domain of a function is important because it clearly defines the set of inputs that the function can accept.
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What happens if a value outside the domain is used in the function?
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If a value outside the domain is used, the function will either produce no output or an output that is not defined by the function's rules.
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Describe an implicit function.
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An implicit function is one where the function is not directly solved for one variable in terms of the others, such as x² + y² = 1.
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How is an inverse function related to the domain and range?
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An inverse function swaps the domain and range of the original function, meaning the output of the original function becomes the input of the inverse.
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Provide an example of a function and identify its domain and range.
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For y = x², the domain is all real numbers, and the range is all non-negative real numbers.
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What is function notation, and how is it used?
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Function notation, such as f(x), is used to denote functions in order to specify the input-output relationship clearly and concisely.
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Why might a function be undefined for certain values?
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A function might be undefined for values that cause division by zero, produce negative square roots, or otherwise fall outside the scope of definition.
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