x2f xx11xf xx1xf0f1f2f x1xx2if x1if x1fyxyyxy1xy1_1FIGURE 150xy=| x |0yFIGURE 16For a more extensive review of absolute values,see Appendix A.

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Chapter A / Exercise 2

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FOUR WAYS TO REPRESENT A FUNCTION17Find a formula for the function graphed in Figure 17.SOLUTIONThe line through and has slope and -intercept , soits equation is . Thus, for the part of the graph of that joins to , wehaveThe line through and has slope , so its point-slope form isSo we haveWe also see that the graph of coincides with the -axis for . Putting this infor-mation together, we have the following three-piece formula for :In Example C at the beginning of this section we considered the costof mailing a large envelope with weight . In effect, this is a piecewise defined functionbecause, from the table of values on page 13, we haveThe graph is shown in Figure 18. You can see why functions similar to this one arecalledstep functions—they jump from one value to the next. Such functions will bestudied in Chapter 2.SymmetryIf a function satisfies for every number in its domain, then is called aneven function. For instance, the function is even becauseThe geometric significance of an even function is that its graph is symmetric with respect EXAMPLE 9fEXAMPLE 10fxx2x2f xf xx2fxfxf xf0.881.051.221.39if 0w1if 1w2if 2w3if 3w4CwwCwf xx2x0if 0x1if 1x2if x2fx2xfif 1x2f x2xy2xory01x2m12, 01, 1if 0x1f xx1, 10, 0fyxb0ym11, 10, 0FIGURE 17xy011FIGURE 18C0.501.001.50012354wPoint-slope form of the equation of a line:See Appendix B.yy1m xx1

18CHAPTER 1to the -axis (see Figure 19). This means that if we have plotted the graph of for ,we obtain the entire graph simply by reflecting this portion about the -axis.If satisfies for every number in its domain, then is called anoddfunction. For example, the function is odd becauseThe graph of an odd function is symmetric about the origin (see Figure 20). If we alreadyhave the graph of for , we can obtain the entire graph by rotating this portionthrough about the origin.Determine whether each of the following functions is even, odd, or neither even nor odd.(a) (b) (c) SOLUTION(a)Therefore is an odd function.(b)So is even.(c)Since and , we conclude that is neither even nor odd.The graphs of the functions in Example 11 are shown in Figure 21. Notice that thegraph of his symmetric neither about the y-axis nor about the origin.FIGURE 2111xyh11yxg1_11yxf_1(a)(b)(c)vEXAMPLE 11hhxh xhxh xhx2xx22xx2ttx1x41x4txff xx5xx5xfxx5x15x5xh x2xx2tx1x4f xx5x180x0ffxx3x3f xf xx3fxfxf xf0x_xƒFIGURE 20An odd functionxy0x_xf(_x)ƒAn even function xFIGURE 19yyx0fy

FOUR WAYS TO REPRESENT A FUNCTION19