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(复习课)概念指数函数对数函数幂函数xayxyalogαxy10,aaR=,0a,0a1.指数与指数幂的运算一.根式:na1.当n为奇数,x=如果nxannana2.当n为偶数,x=na±3.当a=0,即0n00a4.①当n为奇数,a||a②当n为偶数,nnaaa指数运算:nmnmaannaa1srsraaarssraa)(32325522515222)23()32(1)32(52323333312*5.025.033)3(rrraaab)(22232)32((1)(2)(3)(4)(5)图象性质yx0y=1(0,1)y=ax(a1)yx(0,1)y=10y=ax(0a1)定义域:值域:恒过点:在R上是单调在R上是单调a10a1R(0,+∞)(0,1),即x=0时,y=1.增函数减函数指数函数的图像及性质当x0时,y1.当x0时,.0y1当x0时,y1;当x0时,0y1。xay当a1时:若x0,则y1;若x0,则0y1。例:151.01501.0当0a1时:若x0,则0y1;若x0,则y1。例:15.001.015.01.0对数性质:(1)负数和零没有对数;(2)1的对数是零:01loga(3)底数的对数是1:1logaa(4)对数恒等式:NaNalognanalog(5)()logayfxx对数运算:NMNMaaaloglog)(logNMNMaaalogloglogMnManaloglogbmnbanamloglog换底公式:abbccalogloglog特别地:当c=b时,有:abbalog1log当c=10时,有:abbalglglog1)53(log5log3log1515153log412log4log12log33332log52log2522log354log25233log14log433lg4lg4log3(1)(3)(2)(4)图象性质a>10<a<1定义域:值域:过定点:在(0,+∞)上是:在(0,+∞)上是对数函数y=logax(a>0,且a≠1)的图象与性质(0,+∞)R(1,0),即当x=1时,y=0增函数减函数yXOx=1(1,0))1(logayxayXOx=1(1,0))10(logayxa当x1时,y0;当0x1时,y0。当0x1时,y0;当x1时,y0。一、函数的定义域,值域1.求下列函数的定义域)3x(lgx5x6y)4()23x(logy)3()35x(logy)2(3)(5xlog1(1)y2)1x(212),54()54,53(]54,53(),2()2,23(]1,2()2,3(2.求下列函数的值域的值域,求函数已知的值域,求函数,已知)4xlog)(2xlog()x(g]8,1[x)5(12141)x(f]23[x)4()2xx3(logy)3()8x(logy)2()3x(log(1)y22xx22222R),3[]2,(二、函数的单调性3.已知函数y=(1-a)x在R上是减函数,则实数a的取值范围是()A(1,+∞)B(0,1)C(-∞,1)D(-1,1)4.已知不等式a2xax-1的解集为{x|x-1},则实数a的取值范围是()A(0,1)B(0,1)∪(1,+∞)C(1,)D(0,+∞)BC)2(logy)4(),2(log(3))21(y)2(,2(1).5221222222xxxxyyxxxx区间求下列函数的单调递增u=g(x)y=f(u)y=f[g(x)]增增增增增减减减减减减增复合函数单调性xu=g(x)y=f(u)分解各自判断复合定义域6.已知y=loga(2-ax)在[0,1]上是x的减函数,则实数a的取值范围是()A(0,1)B(1,2)C(1,+∞)D(2,+∞)B1log2,0aauyaxua在定义域上为增函数,为定义域上的减函数,因此由于uyaxualog,2则令上有意义,函数在又]1,0[解法1解法2上有意义,函数在函数的定义域为]1,0[),a2,(),a2,(]1,0[01a2.2即21a,a02)1(]10[2minauuaxu上为减函数,,在.2a)31,(7.若函数y=-log2(x2-ax-a)在区间上是增函数,则a的取值范围是())2,32-D.(2],2,322(C.),2,322[B.],2,322[A.:)31,(y4)2(222上递增,只要使在要使设aaaxaaxxu上单调递减。在)3-,1(-u0)31()31(3122aaa。的取值范围故所求解得)2,32-[2a2,)31(2aB上为单调增函数。在定义域证明:函数)2(lg)(.82xxxf。的定义域为时,证明R)(0||2R:2xfxxxxx)22()()2(2:,,R,2221212222112121xxxxxxxxxxxx则且设22))(()(2221212121xxxxxxxx]22)(1)[(22212121xxxxxx22)2()2()(222122212121xxxxxxxx2202,02,021221122212121xxxxxxxxxx上是增函数。在故是增函数,R)()()(lg21xfxfxfxy9.设(1)试判定函数f(x)的单调性,并给出证明;(2)解关于x的不等式xxxxf11lg21)(21])21([xxf三、函数的奇偶性的值是那么是奇函数,是偶函数,设ba24)()110lg()(.10xxxbxgaxxf()A.1B.-1C.D.2121是函数)1(log)(.112xxxfa()A.是奇函数,但不是偶函数B.是偶函数,但不是奇函数C.既是奇函数,又是偶函数D.既不是奇函数,又不是偶函数DA的单调性。,并确定试求实数是奇函数已知函数)(a,122)(.13xfaxfx的奇偶性。,试确定不恒为且是偶函数已知函数)(0)(,)0)(()1221()(F.14xfxfxxfxx3)1(),10(11)(f,aaaaxfxx为奇函数。证明的表达式和定义域;求f(x)(2)f(x)(1)12.已知函数y=log2xy=log3xxy21logxy31log四、特有性质指数函数y=ax对数函数y=logax底大图高底大图底在y轴右侧指数函数的底数越大,其图像越在上方在直线x=1右侧,在x轴上下两侧,指数函数的底数越大,其图像越在下方.__________,03log3log.15之间的关系是那么如果a,bbaba1b.a1b,logalog0,0blog1alog13333解法一:不等式即为b.a1,解法二:如图所示那么如果,03log3logba.__________b,a,3log3logba之间的关系是那么思考:如果那么如果,3log03logba那么如果,3log3log0baba11ba0a1b0xyalogxyblog3.a,1||)2[log.16的取值范围求实数成立上恒有,在区间已知函数yxya若a1,则在区间[2,+∞)上,logax1恒成立。210xy1y=log2xy=logax∴1a2。若0a1,则在区间[2,+∞)上,logax-1恒成立。0xy2-1xy21logxyalog1∴a1。21四、综合应用则下列各式中正确的是已知),10(|log|)(.17axxfa()41241.,41312.23141.,41231.fffDfffCfffBfffAB213141)1,0(,212fffff上单调递减,函数在1是则下列不等式中正确的且若已知),()()(,0|,lg|)(.18bfcfafcbaxxf()A.(a-1)(c-1)0B.ac1C.ab=1D.0ac1Dac
本文标题:指对幂函数复习课
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