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Chapter 2: Functions and Linear Functions

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2.1 Intro to Functions Relation: Set of ordered pairs E.g., {(name, height)} {(year, sales)} {{year, Hawaii population)} Domain: Set of all first components E.g., {name} {year} Range: Set of all second components E.g., {height} {sales}

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Functions Function: set of ordered pairs such that each element of the domain corresponds to exactly one element in the range Which is a function? {(0,9.1), (10,6.7), (20,10.7), (30,13.2)} {(1,5), (2,5), (3,7), (4,8)} {{5,1), (5,2), (7,3,), (8,4)}

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Function Notation y = 2x y is a function of x such that y = 2x f(x) = 2x “f of x equals 2x 2 + 1” What is f(1)? f(1) = 2(1) = 3 What is f(3)? f(3) = 2(3) = 19

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Function Notation (cont) Given: f(x) = 2x What is f(-5)? f(-5) = 2(-5) = 51 What is f(a + b)? f(a + b) = 2(a + b) = 2(a 2 + 2ab + b = 2a 2 + 4ab + b 2 + 1

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Function Notation (cont) Given: g(x) = (3x – 1) / (x – 4) What is g(2)? g(2) = (3(2) – 1) / (2 – 4) = 5/(-2) = -5/2 What is g(-3)? g(-3) = (3(-3) – 1) / ((-3) – 4) = (-9 – 1) / (-3 – 4) = -10 / -7 = 10 / 7

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2.2 Graphs of Functions Given: f(x) = 2x + 4 Graph of f(x): Graph of {(x, y) | y = 2x + 4} x y

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Vertical Line Test Is this a function? f(x) g(x)

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Find f(x) from Graph f(3) = ? Domain of f(x) ? Range of f(x) ?

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2.3 Algebra of Functions Given: f(x) = 2x + 1 g(x) = x - 1 What is: (f + g)(x)? (f + g)(x) = f(x) + g(x) = (2x + 1) + (x – 1) = 3x f(x) g(x) (f + g)(x)

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Algebra of Functions (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) (f ∙ g)(x) = f(x) ∙ g(x) (f / g)(x) = f(x) / g(x)

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Your Turn Given: f(x) = x 2 – 3 g(x) = 4x + 5 Find: a)(f – g)(x) b)(f – g)(2) c)(f / g)(x) d)(f / g)(2)

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2.4 Linear Function & Slope Given: y = 2x + 1 When x = 0, y = 1 For every increase in x, y increases by 2x. (0, 1) Δ y = 2 Δ x = 1

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Linear Function & Slope Given: y = mx + b Y-intercept y Slope: m (0, b) Δ y = m Δ x = 1

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Your Turn Sketch the graph of the following equations. 1.4x – 3y = 6 2.3x = 5y – 15 What is the slope of the line determined by the following points 1.(3, 1) & (5, 4) 2.(-6, -3) & (4, - 3)

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2.5 Slope-Intercept Form of Line Given a line: y-intercept = 5 Slope = 3/4 Find the equation of the line (y – 5)/(x – 0) = 3 / 4 4(y – 5) = 3(x) 4y – 20 = 3x 4y = 3x + 20 y = (3/4)x + 5 (0, 5) ΔyΔy ΔxΔx (x, y) (Δy / Δ x) = 3 / 4

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2.5 Point Slope Form for Line Given: Line passes through (1, 2) & (3, 5) Find the equation of the line m = (5 – 2) / (3 – 1) = 3 / 2 (y - 2) / (x – 1) = 3 / 2 2(y – 2) = (3)(x – 1) 2y – 4 = 3x – 3 2y = 3x + 1 y = (3/2)x + 1/2 (1, 2) ΔyΔy ΔxΔx (3, 5) (x, y)

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