# Chapter 2: Functions and Linear Functions

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Chapter 2: Functions and Linear Functions

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}

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)}

Function Notation y = 2x2 + 1 f(x) = 2x2 + 1 What is f(1)?
y is a function of x such that y = 2x2 + 1 f(x) = 2x2 + 1 “f of x equals 2x2 + 1” What is f(1)? f(1) = 2(1) = 3 What is f(3)? f(3) = 2(3) = 19

Function Notation (cont)
Given: f(x) = 2x2 + 1 What is f(-5)? f(-5) = 2(-5) = 51 What is f(a + b)? f(a + b) = 2(a + b) = 2(a2 + 2ab + b = 2a2 + 4ab + b2 + 1

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 / = 10 / 7

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

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

Find f(x) from Graph f(3) = ? Domain of f(x) ? Range of f(x) ? 20 10
-2 -1 1 2 3 4 -10 Domain of f(x) ? Range of f(x) ? -20

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 + g)(x) g(x) f(x)

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

Your Turn Given: f(x) = x2 – 3 g(x) = 4x + 5 Find: (f – g)(x)

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

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

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

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 (x, y) Δy (0, 5) Δx (Δy / Δx) = 3 / 4

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 (x, y) (3, 5) Δy (1, 2) Δx

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