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Vocabulary • • • • • • Dependent Variable Independent Variable Input Output Function Linear Function Definition • Dependent Variable – A variable whose value depends on some other value. – Generally, y is used for the dependent variable. • Independent Variable – A variable that doesn’t depend on any other value. – Generally, x is used for the independent variable. • The value of the dependent variable depends on the value of the independent variable. Independent and Dependent Variables On a graph; the independent variable is on the horizontal or x-axis. the dependent variable is on the vertical or y-axis. y dependent x independent Example: Identify the independent and dependent variables in the situation. A veterinarian must weight an animal before determining the amount of medication. The amount of medication depends on the weight of an animal. Dependent: amount of medication Independent: weight of animal Your Turn: Identify the independent and dependent variable in the situation. A company charges $10 per hour to rent a jackhammer. The cost to rent a jackhammer depends on the length of time it is rented. Dependent variable: cost Independent variable: time Your Turn: Identify the independent and dependent variable in the situation. Camryn buys p pounds of apples at $0.99 per pound. The cost of apples depends on the number of pounds bought. Dependent variable: cost Independent variable: pounds Definition • Input – Values of the independent variable. – x – values – The input is the value substituted into an equation. • Output – Values of the dependent variable. – y – values. – The output is the result of that substitution in an equation. Function • In the last 2 problems you can describe the relationship by saying that the perimeter (dependent variable – y value) is a function of the number of figures (independent variable – x value). • A function is a relationship that pairs each input value with exactly one output value. Function You can think of a function as an input-output machine. input x2 function y = 5x 30 output Helpful Hint There are several different ways to describe the variables of a function. Independent Variable Dependent Variable x-values y-values Input Output Domain Range x f(x) A function is a set of ordered pairs (x, y) so that each x-value corresponds to exactly one y-value. Function Rule Output variable Input variable Some functions can be described by a rule written in words, such as “double a number and then add nine to the result,” or by an equation with two variables. One variable (x) represents the input, and the other variable (y) represents the output. Linear Function • Another method of representing a function is with a graph. • A linear function is a function whose graph is a nonvertical line or part of a nonvertical line. Example: Representing a Linear Function A DVD buyers club charges a $20 membership fee and $15 per DVD purchased. The table below represents this situation. Number of DVDs purchased x 0 1 2 3 4 5 Total cost ($) y 20 35 50 65 80 95 +15 +15 +15 +15 +15 Find the first differences for the total cost. constant linear Since the data shows a ___________ difference the pattern is __________. If a pattern is linear then its graph is a straight _________. line Relations A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain. The set of output values is called the range. Domain & Range Domain is the set of all x values. Range is the set of all y values. Example 1: {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Domain- D: {1, 2} Range- R: {1, 2, 3} Example 2: Find the Domain and Range of the following relation: {(a,1), (b,2), (c,3), (e,2)} Domain: {a, b, c, e} Range: {1, 2, 3} Page 107 (points Every equation has solution points which satisfy the equation). 3x + y = 5 Some solution points: (0, 5), (1, 2), (2, -1), (3, -4) Most equations have infinitely many solution points. Page 111 Ex 3. Determine whether the given ordered pairs are solutions of this equation. (-1, -4) and (7, 5); y = 3x -1 The collection of all solution points is the graph of the equation. 3.3 Functions •A relation as a function provided there is exactly one output for each input. •It is NOT a function if at least one input has more than one output Page 116 In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT Functions (DOMAIN) FUNCTION MACHINE OUTPUT (RANGE) Example 6 Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates). Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1 1. {(2,5) , (3,8) , (4,6) , (7, 20)} 2. {(1,4) , (1,5) , (2,3) , (9, 28)} 3. {(1,0) , (4,0) , (9,0) , (21, 0)} The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117 Use the vertical line test to visually check if the relation is a function. (-3,3) (4,4) (1,1) (1,-2) Function? No, Two points are on The same vertical line. Use the vertical line test to visually check if the relation is a function. (-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line Examples I’m going to show you a series of graphs. Determine whether or not these graphs are functions. You do not need to draw the graphs in your notes. #1 Function? #2 Function? #3 Function? #4 Function? #5 Function? #6 Function? #7 Function? #8 Function? #9 Function? #10 Function? #11 Function? #12 Function? Function Notation f (x ) “f of x” Input = x Output = f(x) = y Before… Now… y = 6 – 3x f(x) = 6 – 3x x y x f(x) -2 12 -2 12 -1 9 -1 9 0 6 0 6 1 3 1 3 2 0 2 0 (x, y) (input, output) (x, f(x))