Relations, Functions and Evaluations By Mr. Porter.

Relations and Functions. Definitions: Relations: A relation is a set of ordered pairs (points), and is usually defined by some property or rule. The Domain of a relation is the set of all first elements of the ordered pairs. The Range of a relation is the set of all second elements of the ordered pairs. A point (x,y) is a relation, with the first element x being the domain and the second element y being the range of the point. Example of a RELATIONS Dave Sally Joe John 30 25 22 Ordered Pairs (Dave, 30) (Sally, 25) (Joe, 22) (John, 30) (Dave, 22) Domain Range

Definitions Functions: A function is a set of ordered pair in which no two ordered pairs have the same x-coordinate or first value. The domain of a function is the set of all x-coordinates of the ordered pair. The range of a function is the set of all y-coordinates of the order pair. Vertical line Test: A relation is a function if and only if there is no vertical line that crosses the graph - curve more than once. Examples using vertical line test for a function. Every Vertical Line cuts the curve once.Therefore, this curve represents a FUNCTION. Every Vertical Line cuts the curve once.Therefore, this curve represents a FUNCTION. A Vertical Line cuts the curve two or more times.Therefore, this curve represents a RELATION.

Exercise 1 Which of the following curves would represent a function or a relation. a) b) c) d) Relation Function Function Function e) f) g) h) Relation Relation Function Function i) Function

Function Evaluation Notation used to represent a function is of the form f(x), g(x), h(x), p(x) ……, or upper-case F(x), H(x), G(x), … Evaluating a function is the process of replacing the variable with a number or another expression, then either evaluating the result numerically or simplifying the resulting express by expanding. Examples 1) If f(x) = x2 - 3x, evaluate: a) f(5) b) f(-2) c) f(a+3) This means that x = (a+3) This means that x = 5 f(x) = x2 - 3x f(x) = x2 - 3x f(x) = x2 - 3x f(5) = (5)2 - 3(5) f(-2) = (-2)2 - 3(-2) f(a+3) = (a+3)2 - 3(a+3) Expand brackets f(5) = 10 f(-2) = 10 f(a+3) = a2 + 6a + 9 - 3a - 9 Simplify = a2 + 3a It is a good idea to place the value in (..)

Exercise: Evaluate the following functions. 1) Given f(x) = 4 – x3 a) find f(2). b) find f(-3). 2) Given g(x) = 5x – x2 a) find g(1). b) find g(-4). 3) Given h(x) = a) find h(1). (exact value) b) find h(-4). (exact value) a) f(2) = -4 a) g(1) = 4 b) f(-3) = 31 b) g(-4) = -36 4) Given f(x) = 3x - 5 a) f(2a) b) f(a - 3) 5) Given h(x) = 2x - x2 a) h(2 - x) b) h(x2-1) a) f(2a) = 6a – 5 a) h(2 - x) = 2x - x2 b) f(a - 3) = 3a – 14 b) h(x2-1) = -x4 + 4x2 - 3

Multi-Functions These function have a domain [ x-value] restriction which select the expression to be used. Example In this case x = 3, we select the expression x2+2 to evaluate Use f(x) = To evaluate a) f(2) b) f(-5) c) f(1) + f(3) + f(-2) x2 + 2 for x > 2 3 – x for 0 ≤ x ≤ 2 | x | for x < 0 In this case x = 1, we select the expression 3 – x to evaluate In this case x = -2, we select the expression | x | to evaluate In this case x = 2, we select the expression 3 – x to evaluate In this case x = -5, we select the expression | x | to evaluate f(2) = 3 – (2) = 1 f(-5) = | -5 | = 5 f(1) + f(3) + f(-2) = (3 - 1)+ (32+2) + | -2 | = 2 + 11 + 2 = 15

Exercise 1) Use f(x) = To evaluate a) f(2) b) f(1) + f(3) + f(-2) x2 for x > 2 3x for 0 ≤ x ≤ 2 5 for x < 0 2) Use g(x) = To evaluate a) g(2) b) g(1) + g(3) – g(-4) x2 + 3 for x > 0 | 3 + x | for x ≤ 0 a) f(2) = 6 b) f(1) + f(3) + f(-2) = 17 a) g(2) = 7 b) g(1) + g(3) – g(-4) = 15