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9/24/2014 What is a relation? Represent relations and functions • A relation is when an input is paired with an output. • Relations can be in the form of ordered pairs, tables, graphs, or mapping diagrams. 2.1 What is input/output? • The input refers to the x value or domain of relation. • The output refers to the y value or range of the relation. Identify the domain and range • (-2, -3), (-1, -1), (1, 3), (2, 2), and (3, 1) • Domain: {-2, -1, 1, 2, 3} • Range: {-3, -1, 3, 2, 1} {-3, -1, 1, 2, 3} • Make a Mapping Diagram -2 -1 1 2 3 -3 -1 1 2 3 1 9/24/2014 Is it a function? What is a function? • A function is a specific kind of relation, therefore all functions are relations. • Functions have exactly one input mapped to an output value. • In other words each x-value has to be unique, no repeating x-values. X-values cannot repeat, • • • • (4, 5), (5, 6), (6, 7), (7, 7) Yes, each x-value is different (-2, 0), (4, 8), (9, -1), (-2, 0) Yes, (-2, 0) is the same point and was accidentally listed twice. • (-3, 9), (-2, 1), (7, 9), (3, 0), (-2, 10) • No, the -2 is paired with more than 1 y-value. X-values cannot repeat,… Is it a function? • Yes, each x-value only has 1 arrow coming from it. • No, the 1 has two arrows coming from it, so the 1 is mapped to two y-values and the 1 repeats • Domain: {-4, -2, 0, 1, -2} {-4, -2, 0, 1} Not a function • Range:x{3, 1,y 3, -2, -4} {-4, -2, 1, 3} -4 3 -2 -4 -4 -4 -2 -2 -2 1 1 0 0 3 3 1 1 -2 2 9/24/2014 What is the vertical line test? • Yes, each x-value is unique. What are linear equations Linear equations can be written in the form mx+b. • This graph does not pass the vertical line test when x = 28. • This graph does pass the vertical line test so it is a function. They cannot look like: They can look like: • y = x2+1, no x2 or y2 • y = 2x + 1 • y = |2x+1|, no absolute values • 2x + 3y = 5 • = 2 + 1, no square roots • = , x can’t be in the denominator y= • ½x+7=y 3 9/24/2014 Graphing Linear equations • Before you start to graph any linear equation, you should solve the equation for y so that it is in y = mx + b form. Examples • Graph: • y = -2x + 1 • Graph: • 2y – 3x = -8 • The b is the y-intercept so you plot the first point on the y-axis on whatever numeric value the b is. • The m is the slope, so from the b-value you count the rise and then run to make the line. Function notation Normal linear equation Function Notation • y = mx + b • f(x) = mx + b • Function notation is just a fancy way of representing the y for all functions. • Solve: y = 3x + 1, when x = 2 • Solve: f(2) = 3x + 1 • So in function notation y is the same thing as f(x) • Solve: y = ½x – 3, when x = 8 • Solve: f(8) = ½x – 3 4 9/24/2014 Determine whether it is a linear function, then solve • f(3) = 2x2 –x +1 • f(5) = 7 – 3x • • • • P(33) = 1 + 0.03d P(33) = 1 + 0.03(33) P(33) = 1+0.99 P(33) = 1.99 • Domain: 0 < d < 35,800 • Range: We need to find P(35,800) to determine the highest y-value • P(35,800) = 1+0.03(35,800) • P(35,800) = 1 + 1074 • P(35,800) =1075 • Range: 1< P(d) < 1075 5