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Functions and Relations Relations Definition: A relation is a set of ordered pairs. Functions Definition: A function is a relation such that for each.

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Presentation on theme: "Functions and Relations Relations Definition: A relation is a set of ordered pairs. Functions Definition: A function is a relation such that for each."— Presentation transcript:

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2 Functions and Relations

3 Relations Definition: A relation is a set of ordered pairs. Functions Definition: A function is a relation such that for each x, there is exactly one corresponding y.y. 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). is a function. not a function. y –2, –1, 1, 3, 4 –3, –1, 1 –2, 4 3, 1, –1 1, 3, –1, 4, –2 2, –1 Examples: 1.{(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)} 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 = { }

4 y x O –6 –4 –2 –6–4–2 y x O –6 –4 –2 –6–4–2 y x O –6 –4 –2 –6–4–2 y x O –6 –4 –2 –6–4–2 y x O –6 –4 –2 –6–4–2 y x O –6 –4 –2 –6–4–2 y x O –6 –4 –2 –6–4–2 y x O –6 –4 –2 –6–4–2 {(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)} 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. Function Not a Function

5 Function Notation Ex. 1: {(1, 2), (2, 4), (3, 6), (4, 8)}Ex. 2: {(0, 0), (1, 1), (2, 4), (3, 9)} f(x) = x 2 But the most conventional way to denote a function (Ex. 2) is: Function-Value Evaluation Ex. 1: f(x) = x 2 + 1a) f(2) = = 5b) f(5) = = 26c) f(abc) = (abc) = a 2 b 2 c Ex. 2: g(t) = t 2 – 2t + 3 a) g(–3) = (–3) 2 – 2(–3) + 3 = 18b) g(2n) = (2n) 2 – 2(2n) + 3 = 4n 2 – 4n + 3 Ex. 3: H(z) = 2z 2 – 5a) H(4) = 2(4) 2 – 5 = 27b) H(z + 1) = 2(z +1) 2 – 5 = 2(z 2 + 2z + 1) – 5 = 2z 2 + 4z – 3 Note: f(x) is read as f of x, and it doesnt mean f times x; if it does, the notation would have been fx. XY xf:f:2x2x 2x2xx f XY xf:f:x2x2 x2x2 x f 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.


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