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Greatest Integer/Absolute Value Functions Students will be able to find greatest integers and absolute values and graph the both functions. Greatest Integer • Another special function that we will be studying is the greatest integer function. The greatest integer function of a real number x, represented by [x], is the greatest integer that is less than or equal to x. • For example: [4.25] = 4 FHS [6] = 6 [5.99] = 5 Functions [-2.3] = -3 2 • Determine whether each statement below is true or false for all real numbers x and y. 1. [x] + [y] = [x + y] if x = 4.2 and y = 3.1, then x + y = 7.3 [4.2] + [3.1] = [7.3] 4 + 3 = 7 Is this correct? if x = 4.7 and y = 3.9, then x + y = 8.6 [4.7] + [3.9] = [8.6] 4 + 3 = 8 Is this correct? FHS Functions 3 Graph The greatest integer function is sometimes called a step function, because of the shape of its graph. y y = [x] 2 Graph y = [x] x -2 FHS Functions 4 Graph What happens when we change the function? First multiply the function by 2. y y = [x] 2 Graph y =2[x] x On calculator: y = 2int(X) FHS -2 Functions 5 Graph What happens when we change the function? Next multiply the independent variable by 2. y y = [x] 2 Graph y =[2x] x On calculator: y = int(2X) FHS -2 Functions 6 Absolute Value • All integers are composed of two parts – the size and the direction. For example, +5 is five units in the positive direction; –5 is five units in the negative direction. • The absolute value {written like this: 5 }of a number gives the size of the number without the direction. For example, 5 = 5 and 5 = 5. The answer is always positive. FHS Functions 7 Absolute Value • Graphing the absolute value function. Graph: y x y 6 4 x -4 -2 0 2 4 FHS y 4 2 0 2 4 2 x -5 5 -2 -4 -6 Functions 8