2024高考一轮复习 第七讲 函数的奇偶性.doc

1142024高考一轮复习第七讲函数的奇偶性一选择题12023高二下河北期末已知119891119909119886119909119886119909且11989131198911则下列各式一定成立的是A11989131198912B11989101198913C11989111198913D1198910119891122022高三上白山已知函数11989111990941199091119909则不等式3A12B21C12D2132023全国乙卷已知1198911199091199091198901199091198901198861199091是偶函数则119886A2B1C1D242023新高考卷若fxxaln2119909121199091为偶函数则aA1B0C12D152023高三下杭州模拟已知函数119891119909119890211990911989021199092则A1198911199091为奇函数B11989111990912为偶函数C1198911199091为奇函数D11989111990912为偶函数62023绵阳模拟设函数119891119909在定义域119877上满足1198911199091198911199090若119891119909在0上是减函数且11989110则不等式119891119890119909A0B101C10D1119890172023临潼模拟函数119891119909是定义在119877上的奇函数且在0上单调递增11989110则不等式1199091198911199091A02B01C02D1282023千阳模拟在下列函数中为偶函数的是214A119891119909119909cos119909B119891119909119909119888119900119904119909C119891119909119897119899119909D11989111990911990992023闵行模拟下列函数中既不是奇函数也不是偶函数的为A1199100B1199101119909C1199101199092D1199102119909102023崇明模拟下列函数中既是定义域内单调递增函数又是奇函数的为A119891119909119905119886119899119909B1198911199091119909C119891119909119909119888119900119904119909D119891119909119890119909119890119909112023江苏会考已知函数119891119909为奇函数且当1199090时119891119909119897119900119892321199091则1198911A1B0C1D2122023邯郸模拟已知函数1198911199091为偶函数且函数119891119909在1上单调递增则关于x的不等式11989112119909A3B3C2D2二填空题132023高三上哈尔滨开学考若偶函数119891119909在0上单调递减且11989110则不等式1198911199092311990930的解集是142023上海市模拟若函数119910119891119909为偶函数且当1199091198911152023全国甲卷若119891119909119909121198861199091199041198941198991199091205872为偶函数则119886162023高二下浙江期中已知函数119892119909119886511990916为奇函数则实数119886三解答题172023高三上哈尔滨开学考已知函数119891119909119898119909111990921199090211990911199091198991199091求实数119898119899的值2若对任意实数119909都有11989111989021199091205821198911198901199090成立求实数120582的取值范围182023高二下湖州期末已知函数119891119909log119886211990921199091198860且11988611求函数119891119909的奇偶性2若关于119909的方程119891119909log119886119909119898有实数解求实数119898的取值范围192023上海卷函数1198911199091199092311988611199091198881199091198861198861198881198773141当1198860是是否存在实数119888使得119891119909为奇函数2函数119891119909的图像过点13且119891119909的图像与119909轴负半轴有两个交点求实数119886的取值范围202023高二下长春期中设1198911199091198971199001198921311198861199091199091为奇函数119886为常数1求119886的值2若11990924不等式11989111990911990913119909119898恒成立求实数119898的取值范围212023高一下安徽期中已知函数11989111990911989711990011989224119909119886119909是偶函数1求实数119886的值2求方程1198911199091199091的实根的个数3若函数1198921199092119891119909与11990911989912119909119899的图象有且只有一个公共点求实数119899的取值范围222023高一上单县期末已知函数119891119909211990911989811990921119909119877是奇函数1求实数119898的值2讨论函数119891119909在23上的单调性并求函数119891119909在23上的最大值和最小值414答案解析部分1答案A知识点函数单调性的性质函数的奇偶性解析解答解119891119909119886119909119886119909119891119909119891119909为偶函数令119905119886119909则1199050又1199101199051119905在01单调递增当0调递增当1198861时119910119886119909是增函数在1199090上有1198861199091119891119909在1199090单调递增119891119909在1199090上单调递增A119891311989121198912A正确B1198910C11989111198911D1198910故答案为A分析先判断119891119909的奇偶性再根据11989131198911判断119891119909的单调性进而判断选项2答案B知识点函数单调性的性质函数的奇偶性奇偶性与单调性的综合解析解答因为11989111990941199091119909的定义域为R且1198911199094119909111990941199091119909119891119909所以函数119891119909为定义在119877上的奇函数当1199090时则11989111990941199091119909411199091因为11991011199091在0上单调递减则119891119909411199091在0上单调增可得119891119909在0上单调减且函数119891119909为定义在119877上连续不断所以119891119909为定义在119877上的增函数且1198913311989133则3可得3所以不等式3514故答案为B分析根据题意分析可得119891119909为定义在119877上的增函数且为奇函数进而根据函数性质解不等式3答案D知识点偶函数函数的奇偶性解析解答1198911199091199091198901199091198901198861199091是偶函数11989111990911989111990911990911989011990911989011988611990911199091198901199091198901198861199091119909119890119909119890119886111990911989011988611990910恒成立119909不恒为011989011990911989011988611199090解得1198862当1198862时定义域为1199090关于原点对称又满足1198911199091198911199090119891119909为偶函数故选D分析根据偶函数定义进行计算再验证4答案B知识点偶函数解析解答根据题意易得函数定义域为21199091211990910即1199091212关于原点对称119891119909为偶函数则有11989111198911即1198861ln131198861ln3解得1198860检验当1198860时有119891119909119909ln2119909121199091119909ln21199091211990911198911199091198860时119891119909为偶函数故选B分析根据偶函数性质在定义域范畴内代值11989111198911即得答案5答案B知识点函数的奇偶性解析解答方法一因为119891119909119890211990911989021199092所以1198911119909119890221199091198902119909119891119909所以函数119891119909关于11990912对称将119891119909的函数图象向左平移12个单位关于119910轴对称即11989111990912为偶函数方法二因为11989111990912119890211990911198902119909111989011989021199091198902119909119909119877则119891119909121198901198902119909119890211990911989111990912所以11989111990912为偶函数614又1198911199091119890211990921198902119909故1198911111989001198902111989021198911111989041198902119890411198902所以11989111119891111198911111989111故1198911199091为非奇非偶函数又11989111990911198902119909211989021199094故1198911111989041198906111989041198906119891111198900119890211198902所以11989111119891111198911111989111故1198911199091为非奇非偶函数又119891119909121198902119909111989021199093故119891112119890311989051119890311989051198911121198901198902119890所以119891112119891112119891112119891112故11989111990912为非奇非偶函数故答案为B分析方法一由式子结构推出函数对称根据图象平移结合奇偶性的性质进行判断方法二根据函数的奇偶性的定义逐项进行判断可得答案6答案A知识点奇偶性与单调性的综合解析解答1198911199091198911199090即119891119909119891119909故函数119891119909在定义域119877上奇函数若119891119909在0上是减函数则119891119909在0上是减函数1198901199090且119891111989110若1198911198901199091解得1199090故不等式119891119890119909故答案为A分析根据题意可得函数119891119909在定义域119877上奇函数进而可得119891119909在0上是减函数根据题意结合单调性解不等式即可7答案D知识点奇偶性与单调性的综合解析解答因为函数119891119909是奇函数且在0上单调递增所以函数119891119909在0上也单调递增又因为11989110所以11989110不等式1199091198911199091011989111990910714即11990900故答案为D分析由已知利用函数的单调性及奇偶性即可求解出答案8答案C知识点函数的奇偶性解析解答对于A函数119891119909119909119888119900119904119909的定义域为119877且119891119909119909cos119909所以119891119909119891119909故函数不为偶函数对于B函数119891119909119909119888119900119904119909的定义域为119877且119891119909119909cos119909所以119891119909119891119909故函数不为偶函数对于C函数119891119909119897119899119909的定义域为00且119891119909ln119909所以119891119909119891119909故函数为偶函数对于D函数119891119909119909的定义域为0不关于原点对称所以函数不为偶函数故答案为C分析由函数奇偶性的性质结合函数奇偶性的判断逐项进行判断可得答案9答案D知识点函数的奇偶性解析解答A定义域为R且1198911199090119891119909则119891119909为偶函数故错误B11990911990901198911199091119909119891119909则119891119909为奇函数故错误C定义域为R且11989111990911990921199092119891119909则119891119909为偶函数故错误D定义域为R且1198911199092119909119891119909119891119909119891119909119891119909则119891119909既不是奇函数也不是偶函数故正确故答案为D分析利用已知条件结合奇函数和偶函数的定义从而判断出既不是奇函数也不是偶函数的函数10答案D知识点奇偶性与单调性的综合814解析解答对于A119891119909tan119909为奇函数是周期函数在定义域内不单调不正确不符合题意对于1198611198911199091119909定义域为00119891119909119891119909所以119891119909为奇函数但在定义域内不单调不符合题意对于C119891119909119909119888119900119904119909119891119909119909119888119900119904119909119909119888119900119904119909119891119909故函数119891119909119909119888119900119904119909不是奇函数不符合题意对于D1198911199091198901199091198901199090是增函数119891119909119890119909119890119909119891119909是奇函数满足题意故答案为D分析求导根据单调性和奇偶性的定义逐项分析11答案A知识点奇函数解析解答因为函数119891119909为奇函数且当1199090时119891119909119897119900119892321199091所以119891111989111198971199001198923211故答案为A分析利用奇函数性质代入数据计算得到答案12答案A知识点奇偶性与单调性的综合解析解答因为1198911199091为偶函数所以1198911199091的图像关于y轴对称则119891119909的图像关于直线1199091对称因为119891119909在1上单调递增所以119891119909在1上单调递减因为11989112119909故答案为A分析利用已知条件结合偶函数的定义和函数的单调性进而得出关于x的不等式119891121199091198917的解集13答案12知识点函数单调性的性质函数的奇偶性其他不等式的解法914解析解答解由已知条件f10可得1198911199092311990931198911因为偶函数119891119909在0上单调递减所以1199092311990931解得11199092故答案为12分析利用函数的单调性和奇偶性转化成1199092311990931解不等式即可求解14答案12知识点奇函数与偶函数的性质解析解答当119909又因为fx为偶函数所以1198911119891112故答案为12分析利用偶函数的定义即可求解15答案2知识点偶函数解析解答11989111990911990912119886119909sin1199092119909211198862119909cos119909119910cos119909为偶函数为使119891119909为偶函数只需119910119909211198862119909为偶函数11988620即1198862故答案为2分析先利用诱导公式化简sin1199092由三角函数部分为偶函数故只需二次函数部分为偶函数从而得出119886的值16答案12知识点奇函数与偶函数的性质解析解答由函数119892119909119886511990916为奇函数得91910即119886511611988651160解得a12经检验符合题意故答案为12分析利用奇函数的性质列方程求解可得实数a的值101417答案1解当x0时11989111990921199091119909119899因为fx为奇函数所以fxfx所以11989111990921199091119909119899又x0时11989111990911989811990911199092所以11989821198992又当x综上119898211989922解因为e2x0ex0原不等式化为211989021199091119890211990922120582119890119909111989011990921205820令1199051198901199091119890119909则t2原不等式进一步化为t2t30在t2上恒成立记gtt2t3t2当12058222时即4时gtming210所以1符合题意当12058222时即119892119905ming12058221205822412058230显然矛盾综上实数的取值范围为1知识点函数的奇偶性二次函数与一元二次不等式的对应关系解析分析1根据函数是奇函数利用奇函数的定义即可求出mn2利用换元法转化成二次函数在在t2上恒成立问题对120582分类讨论当12058222时当12058222时分别求出120582即可求解18答案1解对于函数119891119909有211990921199090则11990921199092所以函数119891119909的定义域为22119891119909log11988621199092119909log11988621199092119909119891119909故函数119891119909为奇函数11142解由119891119909log119886119909119898可得11990911989821199092119909则11989811990911990921199092119909119909241199092119909141199092令119892119909119909141199092其中2因为函数119910119909111991041199092在22上为增函数故函数119892119909在22上为增函数当2因此实数119898的取值范围是2知识点复合函数的单调性函数的奇偶性函数与方程的综合运用解析分析1先求出函数119891119909的定义域观察是否关于原点对称再利用奇偶函数的定义验证即可2先根据方程119891119909log119886119909119898由实数根解得119898119909141199092再构造函数119892119909119909141199092219答案1当a0时此时1198911199091199092119909119888119909119891119909的定义域为119909011989111990911990921199091198881199091199092119909119888119909若此时119891119909为奇函数则119891119909119891119909211990911990920即119891119909119891119909故不存在实数c使得119891119909为奇函数2由函数119891119909的图像过点1331311988611198881119886解得c1令1198911199090则11990923119886111990911199091198860则11990923119886111990910119909119886119891119909的图像与119909轴负半轴有两个交点方程11990923119886111990910在x轴负半轴有两个解3119886124011990911199092311988610解得11988613又119909119886此时11988623119886111988610解得119886121198861综上所述a的取值范围为131212知识点奇函数一元二次方程的根与系数的关系解析分析1由奇函数定义先得出定义域计算119891119909119891119909是否为0即可判断12142有函数交点分析转化成方程根的分析问题即分析分子二次函数部分的根分布情况及考虑分母不为0情况即得答案20答案1解因为1198911199091198971199001198921311198861199091199091为奇函数则119891119909119891119909119897119900119892131119886119909119909111989711990011989213111988611990911990911198971199001198921311198861199091199091111988611990911990911198971199001198921311198861199092111990920则11198861199092111990921所以11988621即1198861当1198861时1198911199091198971199001198921311199091199091119897119900119892131不合题意当1198861时1198911199091198971199001198921311199091199091由111990911990910可得1199091或119909故11988612解由11989111990911990913119909119898可得119897119900119892131119909119909111990913119909119898则119898令11989211990911989711990011989213111990911990911199091311990911990924因为函数11991011199091199091121199091在24上单调递减所以函数1199101198971199001198921311199091199091在24上单调递增所以119892119909在24上单调递增所以119892119909119898119894119899119892211989711990011989213321989所以119898知识点函数单调性的判断与证明奇函数与偶函数的性质解析分析1由奇函数的性质可得fxfx0代入运算后可得1198861代入验证即可求解出119886的值2不等式11989111990911990913119909119898恒成立转化为11989811989211990911989711990011989213111990911990911199091311990911990924先得出119892119909函数的单调性结合函数的单调性求得gxmin即可求解出实数119898的取值范围21答案1解因为函数11989111990911989711990011989224119909119886119909是偶函数所以119891119909119891119909即1198971199001198922411990911988611990911989711990011989224119909119886119909也即11989711990011989224119909119886119909119897119900119892211988641199091119897119900119892241199091199091198971199001198922411990911988611989711990011989221198864119909141199091198861198864119909111198864119909101314因为对定义域内的任意119909上式恒成立所以11988612解由1可知119891119909的解析式为1198911199091198971199001198922411990911199091198971199001198922211990912119909所以11989111990911990911989711990011989222119909121199091199091198971199001198922114119909因为函数1199101198971199001198922114119909在119877上单调递减又141199090所以函数1199101198971199001198922114119909在119877上的值域为0所以方程1198911199091199091的实根的个数为13解由题可知119892119909211990912119909由119909119892119909可得11989912119909119899211990912119909令1199052119909则1199050所以11989912119909119899211990912119909可化为1198992119905211989911990510令函数119904119905119899211990521198991199051当11989920即1198992时21199051011990512舍去当11989920即1198992时119904119905的图象开口向上因为11990401当1198992因为11990401令1205491198992411989920解得119899223又1199050所以对称轴119905119899211989920所以1198992舍去或119899所以119899223综上实数119899的取值范围是1198991198992223知识点复合函数的单调性奇函数与偶函数的性质二次函数的图象二次函数的性质解析分析1由函数11989111990911989711990011989224119909119886119909是偶函数得119891119909119891119909列方程整理可得11989711990011989224119909119886119897119900119892211988641199091求解可得实数119886的值2由1可得函数1198911199091199091198971199001198922114119909分析函数119891119909119909单调性与值域范围得出方程1198911199091199091的实根的个数3由1得119892119909211990912119909联立两函数建立方程令1199052119909则1199050可得11989921199052141411989911990510构造函数119904119905119899211990521198991199051分11989920119899201198992况结合二次函数的性质可求出实数119899的取值范围22答案1解119891119909211990911989811990921119909119877是奇函数所以11989101198980检验知1198980时119891119909211990911990921119909119877是奇函数所以11989802解1199091119909223且11990911198911199091119891119909221199091119909121211990921199092212119909111990922121199092119909121119909121119909221211990911199092111990911199092119909121119909221211990911即111990911199092又1199091211199092210所以119891119909111989111990920即11989111990911198911199092所以函数119891119909211990911990921在23上单调递减所以当1199092时119891119909取得最大值45当1199093时119891119909取得最小值35知识点函数单调性的判断与证明奇函数与偶函数的性质解析分析1由已知结合奇函数的性质f00可求出实数119898的值2先设1199091119909223且1199091性然后结合单调性即可求解出函数119891119909在23上的最大值和最小值