Relations and Functions Activities

Functions:

In functions activities, a function is a rule, which produce an answer f(x) from an input x. The set of inputs x for which the function is defined is called the domain and f(x) (pronounced “f of x”) is the value of f at x. The set of all possible outputs f(x) as x runs over the domain is called the range of the function.

 Relations:

In relations activities, two-place predicates, such as B(x, y): x is the brother of y, play a central role in mathematics.

Activities of relations:

In relations activities, two-place predicates, such as B(x, y): x is the brother of y, play a central role in mathematics. Such predicates can be used to describe many basic concepts. As examples, consider

  • G(x, y) : x is greater than or equal to y which compares the magnitudes of two values.
  • P(x, y) : x has the same parity as y which compares the parity of two integers.
  • S(x, y) : x has square equal to y which relates a value to its square.

In relations activities, two-place predicates are called relations, probably because of examples such as the brother of given above. To be a little more complete about it, if P (x, y) is a two-place predicate, the domain of discourse for x is the set A, and the domain of discourse for y is the set B, then P is called a relation from A to B. When working with relations, some new vocabulary is used. The set A (the domain of discourse for the first variable) is called the domain of the relation, and the set B (the domain of discourse for the second variable) is called the co-domain of the relation. There are several different ways to specify a relation

Example for relations activities:

In relations activities, if A = {1, 2, 3, 4} and B = {a, b, c, d}, then one of many possible relations from A to B would be {(1, b), (2, c), (4, c)}.

If we name this relation R, we will write R = {(1, b), (2, c), (4, c)}. It would be tough to think of a natural verbal description of R. When thinking of a relation, R, as a set of ordered pairs, it is common to write aRb in place of (a, b) `in` R.

Activities of functions:

  • In functions activities, if a function f(x) is given by an expression in the variable x and the domain is not explicitly specified the domain is understood to be the set of all x for which the expression is meaningful. For example, for the function f(x) =`1/x_2` the domain is the set of all nonzero real numbers x (the value f(0) is not defined because we don’t divide by zero) and the range is the set of all positive real numbers (the square of any nonzero number is positive).
  • The domain and range of the square root function `sqrt(x)` is the set of all nonnegative numbers x. The domain of the function y `=` `sqrt(1-(x)^(2))` is the interval [−1, 1] and the range is the interval [0, 1], i.e. `sqrt(1-(x)^(2))` is meaningful only if −1`<=` x `<=` 1 (otherwise the input to the square root function is negative) and 0 `<=` `sqrt(1-(x)^(2) <= 1)` for functions activities.


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