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Functions and their properties
Domain and range of functions, function as a model, interval notation
Domain and range of functions, function as a model, interval notation
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A function from x to y is a special type of relation where each x value has only one possible y-value. It is expressed in the form
where f and x can be replaced by any letters.
A function can be evaluated for specific values of x by plugging the value into the expression of the function.
The domain of a function is the set of possible inputs it can be given.
The "natural" or "largest possible" domain of a function is all the values of x for which the expression f(x) is defined.
The range of a function is the set of possible values it can output.
If the domain of the function is restricted, the range may need to be restricted as a consequence.
The domain and range of functions are commonly intervals of real numbers.
For example, if f(x) is defined for 1<x≤5, we can write the domain
(∈ means "in" or "element of", and R is all real numbers)
We can also use the equivalent interval notation
where, by IB convention, an outward facing [ means that end is not inclusive (1<x) and an inward facing ] means that end is inclusive (x≤5).
Another common interval notation is
where ( indicates a non-inclusive endpoint and ] indicates an inclusive endpoint. In this style, all brackets are inward facing.
These can also be visualized on a number line:
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A function as a model means using a mathematical relationship to represent real-world phenomena.
By assigning input values (independent variables) and calculating corresponding outputs (dependent variables), a function allows us to approximate, describe, or predict patterns, behaviors, or outcomes.