Section 1: The Square Root Function

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 9.1 The Square Root Function 869 Version: Fall 2007 9.1 The Square Root Function In this section we turn our attention to the square root function, the function defined by the equation f ( x ) = x. (1) We begin the section by drawing the graph of the function, then we address the domain and range. After that, we’ll investigate a number of different transformations of the function. The Graph of the Square Root Function Let’s create a table of points that satisfy the equation of the function, then plot the points from the table on a Cartesian coordinate system on graph paper. We’ll continue creating and plotting points until we are convinced of the eventual shape of the graph. We know we cannot take the square root of a negative number. Therefore, we don’t want to put any negative x -values in our table. To further simplify our computations, let’s use numbers whose square root is easily calculated. This brings to mind perfect squares such as 0, 1, 4, 9, and so on. We’ve placed these numbers as x -values in the table in Figure 1 (b), then calculated the square root of each. In Figure 1 (a), you see each of the points from the table plotted as a solid dot. If we continue to add points to the table, plot them, the graph will eventually fill in and take the shape of the solid curve shown in Figure 1 (c). x 10 y 10 x f ( x ) = x 0 0 1 1 4 2 9 3 x 10 y 10 f (a) (b) (c) Figure 1. Creating the graph of f ( x ) = x . The point plotting approach used to draw the graph of f ( x ) = x in Figure 1 is a tested and familiar procedure. However, a more sophisticated approach involves the theory of inverses developed in the previous chapter. In a sense, taking the square root is the “inverse” of squaring. Well, not quite, as the squaring function f ( x ) = x 2 in Figure 2 (a) fails the horizontal line test and is not one-to-one. However, if we limit the domain of the squaring function, then the graph of f ( x ) = x 2 in Figure 2 (b), where x 0 , does pass the horizontal line test and is Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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870 Chapter 9 Radical Functions Version: Fall 2007 one-to-one. Therefore, the graph of f ( x ) = x 2 , x 0 , has an inverse, and the graph of its inverse is found by reflecting the graph of f ( x ) = x 2 , x 0 , across the line y = x (see Figure 2 (c)). x 10 y 10 f x 10 y 10 f x 10 y 10 f f 1 y = x (a) f ( x ) = x 2 . (b) f ( x ) = x 2 , x 0 . (c) Reflecting the graph in (b) across the line y = x produces the graph of f 1 ( x ) = x . Figure 2. Sketching the inverse of f ( x ) = x 2 , x 0 . To find the equation of the inverse, recall that the procedure requires that we switch the roles of x and y , then solve the resulting equation for y . Thus, first write f ( x ) = x 2 , x 0 , in the form y = x 2 , x 0 . Next, switch x and y . x = y 2 , y 0 (2) When we solve this last equation for y , we get two solutions, y = ± x. (3) However, in equation (2) , note that y must be greater than or equal to zero. Hence, we must choose the nonnegative answer in equation (3) , so the inverse of f ( x ) = x 2 , x 0 , has equation f 1 ( x ) = x.
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