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Situation 41: Square Roots Prepared at Penn State Mid-Atlantic Center for Mathematics Teaching and Learning 14 July 2005 – Tracy, Jana, Christa, Jim Prompt A teacher asks her students to sketch the graph of f ( x ) = "x . A student responds, “That’s impossible! You can’t take the square root of a negative number!” ! Commentary Mathematical Foci Mathematical Focus 1 The domain of the square root function is all nonnegative real numbers. To find the domain of f ( x ) = "x , one may algebraically solve the inequality "x # 0 . Mathematical Focus 2 ! ! Use a transformation of the graph of the known function, g( x ) = x , in order to generate a graph of a less familiar function, f ( x ) = "x . If the graph of g( x ) = x is reflected about the vertical axis, the result is the graph of f ( x ) = "x as is shown in the following figure. It is important to recognize that the point (0, 0) is on ! both graphs. ! ! ! SIT_M_050716_sqrt.doc Page 1 of 2 Function Plot no data 6 5 4 3 2 1 0 -1 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 x y = y = x !x Mathematical Focus 3 Verify that the function f ( x ) = "x makes sense by testing a few specific negative values and a few specific positive values for x. It might help to choose numbers whose absolute values are perfect squares, such as these shown on the following chart: ! X "x -4 "("4) = 2 4 "4 is not a real number ! -1 "("1) = 1 ! 1 "1 is not a real number ! 0 "0 = 0 ! The results for x-values –4, 4, -1, 1, and 0 suggest the function’s domain contains ! all non-positive numbers. ! SIT_M_050716_sqrt.doc Page 2 of 2