04Ffuncnotation

= Function Notation =

Recall that
 * a function is any relation where for any x value in the domain, there is only one y value.
 * the domain of a function is the set of all x values for that function (x is the independent variable)
 * the range of a function is the set of all y values for that function (y is the dependent variable)

We often write the rule of a function as y = f(x)
 * this is read as y is a function with the variable being x
 * y = f(3) means make y the result of substituting x = 3 into the rule


 * Maximal Domain **


 * The maximal domain of a function, is the largest domain for which the rule can be defined
 * for example,
 * f(x) = 2x + 3, has a maximal domain of R
 * f(x) = sqrt(x) has a maximal domain of x __>__ 0
 * If a function is stated without any domain given, then it is assumed to have its maximal domain


 * Formal Definition of a Function **

The formal definition of a function looks like this:

... ... ... f: D —> CD, f(x) = {rule}

... ... where
 * f is the name of the function. Standard names for functions are f, g, h
 * D is the domain of the function, written in interval notation
 * CD is the Co-Domain, which is the set of values from which the range will be selected by the rule
 * The Co-Domain is almost always R (the set of Real numbers)
 * The arrow --> is read as "maps onto",
 * ie the values of the domain are mapped onto (or matched up with) values from the co-domain
 * f(x) = {rule} is where you write the rule for this function.

Examples of formally defined functions

... ... ... f: R —> R, f(x) = 2x + 3

... ... ... g: [0, 10) —> R,g(x) = x 2 + 2

... ... ... h: R + —> R, h(x) = sin(x)

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