Relations & Functions

Transcription

1 Relations & Functions A RELATION is a set of ordered pairs. A relation may be designated in several ways : 1. If a relation is a small finite set of ordered pairs, it may be shown in : a. Roster Notation : {(a, 2), (b, 1), (c, ), (c, 3)}. b. Table Form : as per the picture on the right. 2. A graph in the (xy coordinate) plane : as per the graph of the circle : Note : Any graph in the plane designates a relation. 3. An equation in two variables : (x 3) 2 y2 = 1 Note : The set of all ordered pairs that satisfies the equation is the relation. The DOMAIN of a relation is the set of all 1 st coordinates of its ordered pairs. The RANGE of a relation is the set of all 2 nd coordinates of its ordered pairs. Thus is 1. a. above (roster notation), the Domain is {a, b, c}, and the Range is {1, 2, 3, }. In 1. b., the Domain is {0, 2,, 6, 8, 10}, and the Range is { 1, 1, 5}. In 2., the Domain is the set {x x }, and the Range is {y y }. The Domain and Range of 3. is a bit harder to determine. We ll save that for later. A FUNCTION is a relation in which no two distinct ordered pairs have the same first coordinate. Relation 1. a. fails to be a function: the value c appears as the 1 st coordinate of two distinct ordered pairs. Relation 1. b. satisfies the definition of being a function. The fact that 1 appears as the 2 nd coordinate of more than one ordered pair does not violate the definition of function. Relation 2. fails to be a function. Note that both (0, ) and (0, ) are on the graph. Relation 3. fails to be a function. The ordered pairs (8, ) and (8, ) both satisfy the equation. 1

2 The Vertical Line Test Given the graph of a relation, if any vertical line intersects the graph at more than one point, the relation is not a function. Example 1 : The following are functions : (x + 5) 2 + y 3 = x y = (x 2)3 + 2 The following are not functions : y 2 x 2 = 10 x = y In a function, the variable representing the domain values is called the INDEPENDENT VARIABLE. The variable representing the range values is called the DEPENDENT VARIABLE. If a relation is given by an equation, and the equation can be solved for the dependent variable, then the equation represents a function. For example : 3x y = 12 can be solved for y = 3 x 3. Thus y is said to be a function of x. (x 3) 2 y2 = 1 cannot be solved for y without using ± notation, which represents two values, not one. 2

3 Function Notation If y is a function of x, we write y = f(x), which reads : y equals f of x. Note well : f(x) DOES NOT mean f times x. The notation f(x) cannot be separated. It exists as a single entity. So we can write a function such as y = 3 x 3 using function notation : f(x) = 3 x 3. The function is f, whereas f(x) is the value of f given the value x. We can think of a function machine, where the independent variable values are the inputs of the machine, and the dependent variable values are the outputs of the machine. In the equation f(x) = 3 x 3, x represents the input, and f(x) represents the output. The GRAPH of an equation in two variables is the set of all ordered pairs that satisfy the equation. Then the GRAPH of a function is the set of ordered pairs ( x, f(x) ). Given a function such as f(x) = 2x 2 2x 2, the symbol inside the parentheses, in this case x, is called the ARGUMENT of the function. We can EVALUATE a function for a given value by substituting the given value for the argument of the function, wherever it occurs. For example, Evaluate the function f(x) = 2x 2 2x 2 for x = 2. We find f of negative two : f( 2) = 2( 2) 2 2( 2) 2 = 2() + 2 = 6. Thus f( 2) = 6. A function can be denoted by letters other than f : Evaluate the function g(x) = x 2 for x = g() = () 2 = 2. So g() = 2. Find x such that g(x) =. In this case, we take the expression that defines g(x), which is x 2, and set that equal to : x 2 = x = 6. 3

4 There are some Basic Functions which form the bases of others : CONSTANT Function IDENTITY Function SQUARE Function f(x) = 2 f(x) = x f(x) = x 2 Dom = R Rng = {2} Dom = R Rng = R Dom = R Rng = { x x 0 } CUBIC Function ABSOLUTE VALUE Function RECIPROCAL Function f(x) = x 3 f(x) = x f(x) = 1 x Dom = R Rng = R Dom = R Rng = { y y 0 } Dom = { x x 0 } Rng = { y y 0 } SQUARE ROOT Function f(x) = x CUBE ROOT Function 3 f(x) = x Dom = R* Rng = R* Dom = R Rng = R

5 Reasons why a particular Relation could fail to be a Function: (If possible, give example) i) One or more domain elements is related to more than one range element, or Two or more ordered distinct ordered pairs share (have) the same 1 st coordinate. ii) (The relation is given as a graph and) it fails the vertical line test. That is, there is a vertical line that intersects the graph of the relation at more than one point. Reasons why a particular Relation is a Function : i) No two distinct ordered pairs have the same first coordinate. ii) No vertical line intersects the graph of the relation at more than one point. Problems : A. Relations : Domain, Range, Function Testing 1. Let R = {(a, 1), (b, 2), (c, 3), (d, )}. a) Write the Domain of R in set notation. b) Write the Range of R in set notation. c) Is the relation a function? Give reason. 2. Let the Domain of relation P be the set of states in the U.S., and let the 2 nd coordinate of a particular ordered pair be the Capital of that state. a) Is the relation a function? 3. Let S = {(1, 3), (3, 1), (2, ), (, 2), (3, 5), (5, 3), (, 6), (6, ), }. a) What is the Domain of the Relation? b) What is the Range of the Relation? c) Is the Relation a Function? Give reason. B. Functions : Domain, Range, Values 1. Let f(x) = x 2 a) What is f(0)? b) What is f( 2)? c) What is f(2)? d) For what value of x is f(x) = 5? e) What is the Domain of f? f) What is the Range of f? 2. Let g(x) = 1 x. a) What is the Domain of g? b) What is the Range of g? c) For what value of x is g(x) = π? 5