Math 213, Section B1, Quiz 2 (Solutions); Friday, Jan 25, 2008
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Let f:RโRbe given by f(x) = exโ1.
(1) Is the function fone-to-one?
(2) Find the range of f.
(3) Is fsurjective?
(4) For g:RโR,g(x) = x2+x, compute the functions fโฆgand gโฆf.
Solution.
(1) The function f(x) is one-to-one. Indeed, if f(x1) = f(x2) for some
x1=x2โRthen ex1โ1 = ex2โ1. Hence ex1=ex2and x1=x2.
(2) The range of fis (โ1,โ) = {yโR|y > โ1}.
Indeed, by definition,
range(f) = {f(x)|xโR}={exโ1|xโR}=
{yโR|y=exโ1 for some xโR}={yโR|yโ1 = exfor some xโR}=
={yโR|yโ1>0}={yโR|y > โ1}.
(3) No, the function fis not surjective. For example, the number โ3
belongs to the co-domain of f(which, by definition of f, is the set R), but
โ36โ (โ1,โ) = range(f).
(4) We have
(fโฆg)(x) = f(g(x)) = f(x2+x) = ex2+xโ1,
(gโฆf)(x) = g(f(x)) = g(exโ1) = (exโ1)2+exโ1 = e2xโex.
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