REPRESENTATION OF FUNCTION, Study notes of Mathematics

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REPRESENTATION OF FUNCTION

The list and the set of ordered pairs below show the same relation. Each letter is paired with a number. Letter Number I 4 L 5 O 6 V 8 E 3 M 6 A 2 T 8 H 4 {(I,4), (L,5), (O,6), (V,8), (E,3), (M,6), (A,2), (T,8), (H,4)} The domain is {I, L, O, V, E, M, A, T, H} and the range is {2, 3, 4, 5, 6, 8}. Note that for each letter corresponds exactly one number. This is a special kind of relation called function. The members of the domain can be called inputs and the members of the range can be called outputs. Arrows can be used to describe correspondence in the function. Domain I L O V E M A T H Range 2 3 4 5 6 8 {(I,4), (L,5), (O,6), (V,8), (E,3), (M,6), (A,2), (T,8), (H,4)} A relation is a set of ordered pairs. The domain of a relation is the set of first coordinates. The range is the set of second coordinates. A function is a relation in which each element of the document corresponds to exactly one element of the range.

Input Output 1 2 2 4 3 6 4 8 CHARACTERISTICS OF A FUNCTION

1. Each element in domain X must be matched with exactly one element in range Y.
2. Some elements in Y may not be matched with any element in X.
3. Two or more elements in X may be matched with the same element in Y. OTHER WAYS TO DESCRIBE THE FUNCTION
4. MAPPING DIAGRAM Input (X) Output (Y) 1 2 2 5 3 10 4 14
5. TABLE OF VALUES Input (X) 1 2 3 4 Output (Y) 2 5 10 14 If we are given a set of ordered pairs, we can easily determine whether the relation is a function or not by simply looking if each first element is used only once in the given set.

3. GRAPH

4. RULE OR CORRESPONDENCE

f: x x^2 + 1, x = 1, 2, 3, 4 Notice that the set of ordered pairs of numbers, the mapping diagram, the table of values, and the graph clearly show that each value of Y is obtained by adding 1 to the square of X. Hence, this is the rule or correspondence, expressed in words for the said relation.

5. Equation: The rule or correspondence can be described by the equation y = x^2 + 1. You have seen that not every set of ordered pairs defines a function. Similarly, not all equations with the variables x and y define a function. If an equation is solved for y and more than one value of y can be obtained for a particular value of x, then the equation does not define y as a function of x. 2 4 6 8 10 12 14 (^1 2 3 )     _Note: When finding the domain and range of a function involving:
6. A radical with an even index: Radicand must be non-negative. Hence, the radicand must be greater than or equal to zero.
7. A fraction: Denominator must not be equal to zero._