f(x) = 0if0x <13,f(x) = 1if13x <23,f(2) = 2if23x <1;moreover,f(x) =−3if−1x <−23,f(x) =−2if−23x <−13,f(x) =−1if−13x <0.In general, ifmis any integer, thenf(x) = 3mifmx < m+13,f(x) = 3m+ 1ifm+13x < m+23,f(x) = 3m+ 2ifm+23x < m+ 1.Because every integer is equal to 3mor to 3m+ 1 or to 3m+ 2 for some integerm, we see that the rangeoffincludes the setZof all integers. Becausefcan assume no values other than integers, we can concludethat the range offis exactlyZ.C01S01.019:Givenf(x) = (−1)[[x]], we first note that the values of the exponent [[x]] consist of all theintegers and no other numbers. So all that matters about the exponent is whether it is an even integer oran odd integer, for if even thenf(x) = 1 and if odd thenf(x) =−1. No other values off(x) are possible,so the range offis the set consisting of the two numbers−1 and 1.C01S01.020:If 0< x1, thenf(x) = 34. If 1< x2 thenf(x) = 34 + 21 = 55. If 2< x3 thenf(x) = 34+2·21 = 76. We continue in this way and conclude with the observation that if 11
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