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Functions and Relations

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Relations Functions Definition: Definition: A relation is a set of ordered pairs. A function is a relation such that for each x, there is exactly one corresponding y. Examples: {(1, –1), (2, 1), (3, 3), (4, 5)} 2. {(–2, 3), (–1, 1), (1, –1), (3, 1), (4, 3)} 3. {(–3, 1), (–1, 3), (–1, –1), (1, 4), (1, –2)} 4. {(–2, 2), (–2, –1), (4, 2), (4, –1)} is a function. a function. not a function. not a function. Definitions: Domain is the set of all possible values of x of a function (or a relation). Range is the set of all possible values of __ of a function (or a relation). y What are the domain and range of the four examples above? 1. Domain = {1, 2, 3, 4} Range = {–1, 1, 3, 5} 2. Domain = { } Range = { } 3. Domain = { } Range = { } 4. Domain = { } Range = { } –2, –1, 1, 3, 4 3, 1, –1 –3, –1, 1 1, 3, –1, 4, –2 –2, 4 2, –1

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**The graph of a relation is a function if and only if**

{(1, –1), (2, 1), (3, 3), (4, 5)} {(–2, 3), (–1, 1), (1, –1), (3, 1), (4, 3)} {(–3, 1), (–1, 3), (–1, –1), (1, 4), (1, –2)} {(–2, 2), (–2, –1), (4, 2), (4, –1)} y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 Function Function Not a Function Not a Function y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 The Vertical Line Test: The graph of a relation is a function if and only if every vertical line drawn only intersects the curve in at most one point.

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**X Y X Y Function Notation Ex. 1: {(1, 2), (2, 4), (3, 6), (4, 8)}**

1 2 3 4 9 x f: 2x x f: x2 2x x f x2 x f Note: f(x) is read as “f of x”, and it doesn’t mean f times x; if it does, the notation would have been fx. But the most conventional way to denote a function (Ex. 2) is: f(x) = x2 Name of the function Name of the variable (i.e., the input) Definition of the function; here, the definition is: no matter what the input is, the output will be the square of the input. Function-Value Evaluation Ex. 1: f(x) = x2 + 1 a) f(2) = = 5 b) f(5) = = 26 c) f(abc) = (abc)2 + 1 = a2b2c2 + 1 Ex. 2: g(t) = t2 – 2t + 3 a) g(–3) = (–3)2 – 2(–3) + 3 = 18 b) g(2n) = (2n)2 – 2(2n) + 3 = 4n2 – 4n + 3 Ex. 3: H(z) = 2z2 – 5 a) H(4) = 2(4)2 – 5 = 27 b) H(z + 1) = 2(z +1)2 – 5 = 2(z2 + 2z + 1) – = 2z2 + 4z – 3

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