Function: Domain and Range, Methods of Representation
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What is a function in mathematics?
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.
What is the domain of a function?
The domain of a function is the complete set of possible input values (x-values) which produce a valid output from the function.
What is the range of a function?
The range of a function is the set of all possible output values (y-values), which result from using the function.
How is a function typically represented?
A function can be represented as an equation, a graph, a table of values, or a mapping diagram.
Explain one way to represent a function graphically.
Graphically, a function is represented by plotting it on a coordinate plane, typically as a line or curve.
What is an explicit equation in the context of a function?
An explicit equation is a mathematical expression that clearly defines the relationship between variables, generally written as y = f(x).
How can a table be used to represent a function?
A table represents a function by listing input values (domain) alongside their corresponding output values (range).
What is a mapping diagram?
A mapping diagram depicts a function by showing how each element of the domain is related to an element in the range using arrows.
Why is it important to specify the domain of a function?
Specifying the domain of a function is important because it clearly defines the set of inputs that the function can accept.
What happens if a value outside the domain is used in the function?
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.
Describe an implicit function.
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.
How is an inverse function related to the domain and range?
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.
Provide an example of a function and identify its domain and range.
For y = x², the domain is all real numbers, and the range is all non-negative real numbers.
What is function notation, and how is it used?
Function notation, such as f(x), is used to denote functions in order to specify the input-output relationship clearly and concisely.
Why might a function be undefined for certain values?
A function might be undefined for values that cause division by zero, produce negative square roots, or otherwise fall outside the scope of definition.