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Lesson 2.1: Domains, Ranges, and Compositions of Functions A relation is a relationship between two variables that results in a set of ordered pairs. A is a special kind of relation: it is a relationship between two variables such that each value of the first variable is paired with value of the second variable. The of a function is the set of all possible values of the first variable. The of a function is the set of all possible values of the second variable. You can also use the to determine if a graph represents a function: if every vertical line intersects a given graph at no more than one point, then the graph represents a function. e.g. Which of the following relations are functions? Explain. a) 2,3 , 4,1 , 0,1 , 1, 2 b) c) 5, 2 , 1,3 , 0, 4 , 5,1 d) Relations and functions can also be described using mapping diagrams that illustrate the pairing between each domain element and its corresponding range element(s). For example, the relation 5, 2 , 1,3 , 0, 4 , 5,1 could be described as: 5 -2 3 1 0 -4 1 If there is a correspondence between values of the domain, x , and values of the range, y , that is a function, then y f x , and x, y can be written as x, f x . The notations f x and f : x are “. The number represented by f x is the value of the function f at x . For read as “ example, a function f that is mapping any domain element x to a range element that is equal to 2x 1 can be notated as either f x 2 x 1 or f : x 2 x 1 . The variable x is known as the variable, while the variable y , or f x , is known as the variable. e.g. Given the function g : x x 2 3x , evaluate: a) g 1 c) g 2a 1 b) g : 2 e.g. Given the function h x 1 4 x , find x when: b) h x 3a 7 a) h : x 9 e.g. Which of the following relations are functions? Explain. a) b) 5 -2 5 3 1 0 c) y x 2 1 -4 1 0 d) y 1 3 x 2x 3 2 -2 3 Lesson 2.1 (continued) Many relations have restricted domains and/or ranges that can be expressed in set or interval notation. A collection of individual points is known as a relation, while a line, ray, segment, or smooth curve is known as a relation. e.g. State the domain and range for each of the following relations: a) b) 5 -2 3 1 -4 1 0 c) 0,3 , 2, 4 , 2,1 , 3,3 d) f x 1 x3 2 When you apply a function rule onto the result of another function rule, you are performing a of functions. The composition of f with g , denoted f g or f g x , is defined by f g x . The domain of y f g x is the set of domain values of g whose range values are in the domain of f . The function f g is called the composite function of f with g . To evaluate a composition of functions such as f g 1 , you have a choice of two methods: 1. you could start by evaluating g 1 , and then substitute the result into the function f , OR 2. you could start by finding an expression for f g x , and then substitute x 1 into this composite function e.g. Given f x 4 x 3 and g x x 2 1 , find: a) f g 3 c) f f e) f g f Note from the last two examples above that f compositions of functions are not usually b) g f 2 d) g f a 1 f) g f g g f . In fact, this is usually the case meaning that . However, there are some exceptions. e.g. Find two functions f and g whose composite functions f g and g f are equivalent.