第10講指數(shù)與指數(shù)函數(shù)（精講）題型目錄一覽①指數(shù)冪的化簡與求值②指數(shù)函數(shù)的圖像與性質(zhì)③解指數(shù)方程與不等式④指數(shù)函數(shù)的綜合應(yīng)用★【文末附錄-指數(shù)運(yùn)算和指數(shù)函數(shù)思維導(dǎo)圖】一、知識點(diǎn)梳理一、知識點(diǎn)梳理1.指數(shù)及指數(shù)運(yùn)算(1)根式的定義：一般地，如果SKIPIF1<0，那么SKIPIF1<0叫做SKIPIF1<0的SKIPIF1<0次方根，其中SKIPIF1<0，SKIPIF1<0，記為SKIPIF1<0，SKIPIF1<0稱為根指數(shù)，SKIPIF1<0稱為根底數(shù)．(2)根式的性質(zhì)：當(dāng)SKIPIF1<0為奇數(shù)時，正數(shù)的SKIPIF1<0次方根是一個正數(shù)，負(fù)數(shù)的SKIPIF1<0次方根是一個負(fù)數(shù).當(dāng)SKIPIF1<0為偶數(shù)時，正數(shù)的SKIPIF1<0次方根有兩個，它們互為相反數(shù).(3)指數(shù)的概念：指數(shù)是冪運(yùn)算SKIPIF1<0中的一個參數(shù)，SKIPIF1<0為底數(shù)，SKIPIF1<0為指數(shù)，指數(shù)位于底數(shù)的右上角，冪運(yùn)算表示指數(shù)個底數(shù)相乘.(4)有理數(shù)指數(shù)冪的分類①正整數(shù)指數(shù)冪SKIPIF1<0；②零指數(shù)冪SKIPIF1<0；③負(fù)整數(shù)指數(shù)冪SKIPIF1<0，SKIPIF1<0；④SKIPIF1<0的正分?jǐn)?shù)指數(shù)冪等于SKIPIF1<0，SKIPIF1<0的負(fù)分?jǐn)?shù)指數(shù)冪沒有意義．(5)有理數(shù)指數(shù)冪的性質(zhì)①SKIPIF1<0，SKIPIF1<0，SKIPIF1<0；②SKIPIF1<0，SKIPIF1<0，SKIPIF1<0；③SKIPIF1<0，SKIPIF1<0，SKIPIF1<0；④SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．2.指數(shù)函數(shù)SKIPIF1<0SKIPIF1<0SKIPIF1<0圖象性質(zhì)①定義域SKIPIF1<0，值域SKIPIF1<0②SKIPIF1<0，即時SKIPIF1<0，SKIPIF1<0，圖象都經(jīng)過SKIPIF1<0點(diǎn)③SKIPIF1<0，即SKIPIF1<0時，SKIPIF1<0等于底數(shù)SKIPIF1<0④在定義域上是單調(diào)減函數(shù)在定義域上是單調(diào)增函數(shù)⑤SKIPIF1<0時，SKIPIF1<0；SKIPIF1<0時，SKIPIF1<0SKIPIF1<0時，SKIPIF1<0；SKIPIF1<0時，SKIPIF1<0⑥既不是奇函數(shù)，也不是偶函數(shù)【常用結(jié)論】1.指數(shù)函數(shù)常用技巧(1)當(dāng)?shù)讛?shù)大小不定時，必須分“SKIPIF1<0”和“SKIPIF1<0”兩種情形討論．(2)當(dāng)SKIPIF1<0時，SKIPIF1<0，SKIPIF1<0；SKIPIF1<0的值越小，圖象越靠近SKIPIF1<0軸，遞減的速度越快．當(dāng)SKIPIF1<0時SKIPIF1<0，SKIPIF1<0；SKIPIF1<0的值越大，圖象越靠近SKIPIF1<0軸，遞增速度越快．(3)指數(shù)函數(shù)SKIPIF1<0與SKIPIF1<0的圖象關(guān)于SKIPIF1<0軸對稱．二、題型分類精講二、題型分類精講刷真題明導(dǎo)向刷真題明導(dǎo)向一、單選題1．已知函數(shù)SKIPIF1<0，則對任意實(shí)數(shù)x，有（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．設(shè)SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．已知函數(shù)SKIPIF1<0是偶函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0，則該函數(shù)在SKIPIF1<0上的圖像大致是（

）A． B．C． D．4．下列函數(shù)中最小值為4的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．已知SKIPIF1<0，則SKIPIF1<0（

）A．25 B．5 C．SKIPIF1<0 D．SKIPIF1<06．若SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．設(shè)SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型一指數(shù)冪的化簡與求值策略方法指數(shù)冪運(yùn)算的一般原則(1)有括號的先算括號里的，無括號的先算指數(shù)運(yùn)算．(2)先乘除后加減，負(fù)指數(shù)冪化成正指數(shù)冪的倒數(shù)．(3)底數(shù)是負(fù)數(shù)，先確定符號；底數(shù)是小數(shù)，先化成分?jǐn)?shù)；底數(shù)是帶分?jǐn)?shù)的，先化成假分?jǐn)?shù)．(4)若是根式，應(yīng)化為分?jǐn)?shù)指數(shù)冪，盡可能用冪的形式表示，運(yùn)用指數(shù)冪的運(yùn)算性質(zhì)來解答．【典例1】計算：(1)SKIPIF1<0；(2)已知：SKIPIF1<0，求SKIPIF1<0的值.【題型訓(xùn)練】一、單選題1．SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．下列結(jié)論中，正確的是（

）A．設(shè)SKIPIF1<0則SKIPIF1<0 B．若SKIPIF1<0，則SKIPIF1<0SKIPIF1<0C．若SKIPIF1<0，則SKIPIF1<0 D．SKIPIF1<0二、填空題3．若SKIPIF1<0，則SKIPIF1<0______4．已知SKIPIF1<0，化簡二次根式SKIPIF1<0的值是________5．已知SKIPIF1<0，則SKIPIF1<0=__________6．已知SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的值為__________．三、解答題7．（1）計算SKIPIF1<0；（2）若SKIPIF1<0，求SKIPIF1<0的值．8．（1）計算：SKIPIF1<0；（2）已知SKIPIF1<0是方程SKIPIF1<0的兩根，求SKIPIF1<0的值.題型二指數(shù)函數(shù)的圖像與性質(zhì)策略方法解決指數(shù)函數(shù)有關(guān)問題，思路是從它們的圖像與性質(zhì)考慮，按照數(shù)形結(jié)合的思路分析，從圖像與性質(zhì)找到解題的突破口，但要注意底數(shù)對問題的影響.【典例1】函數(shù)SKIPIF1<0有兩個不同的零點(diǎn)，則SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）的圖象可能為（

）A．B．C．D．【典例2】已知函數(shù)SKIPIF1<0的圖像恒過一點(diǎn)P，且點(diǎn)P在直線SKIPIF1<0的圖像上，則SKIPIF1<0的最小值為（）A．4 B．6 C．7 D．8【典例3】比較下列幾組值的大?。?1)SKIPIF1<0和SKIPIF1<0；(2)SKIPIF1<0和SKIPIF1<0；(3)SKIPIF1<0和SKIPIF1<0；(4)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．【題型訓(xùn)練】一、單選題1．如圖中，①②③④中不屬于函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0中一個的是（

）A．① B．② C．③ D．④2．函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）與函數(shù)SKIPIF1<0的圖象可能是（

）A． B．C． D．3．函數(shù)SKIPIF1<0（其中SKIPIF1<0，SKIPIF1<0）的圖象恒過的定點(diǎn)是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）的圖象過定點(diǎn)SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．函數(shù)SKIPIF1<0的圖像恒過定點(diǎn)A，若點(diǎn)A在雙曲線SKIPIF1<0上，則m-n的最大值為（

）A．6 B．-2 C．1 D．46．已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．設(shè)函數(shù)SKIPIF1<0，若SKIPIF1<0為增函數(shù)，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．已知SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0二、多選題9．點(diǎn)SKIPIF1<0在函數(shù)SKIPIF1<0的圖象上，當(dāng)SKIPIF1<0，則SKIPIF1<0可能等于（

）A．-1 B．SKIPIF1<0 C．SKIPIF1<0 D．0三、填空題10．請寫出一個同時滿足下列條件①②③的函數(shù)SKIPIF1<0____________．①SKIPIF1<0；②對任意SKIPIF1<0，當(dāng)SKIPIF1<0時，SKIPIF1<0；③SKIPIF1<0．11．已知SKIPIF1<0為SKIPIF1<0上的奇函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0，則不等式SKIPIF1<0的解集為___________.12．若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，則k的取值范圍為____________．四、解答題13．已知函數(shù)SKIPIF1<0（a為常數(shù)）和函數(shù)SKIPIF1<0，且SKIPIF1<0為奇函數(shù)．(1)求實(shí)數(shù)a的值；(2)設(shè)不等式SKIPIF1<0恒成立，試求實(shí)數(shù)SKIPIF1<0的范圍．題型三解指數(shù)方程與不等式策略方法指數(shù)方程或不等式的解法(1)解指數(shù)方程或不等式的依據(jù)①af(x)＝ag(x)?f(x)＝g(x)．②af(x)＞ag(x)，當(dāng)a＞1時，等價于f(x)＞g(x)；當(dāng)0＜a＜1時，等價于f(x)＜g(x)．(2)解指數(shù)方程或不等式的方法先利用冪的運(yùn)算性質(zhì)化為同底數(shù)冪，再利用函數(shù)單調(diào)性轉(zhuǎn)化為一般不等式求解．【典例1】不等式SKIPIF1<0對于SKIPIF1<0恒成立，則SKIPIF1<0的取值范圍是______．【題型訓(xùn)練】一、單選題1．已知集合SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．設(shè)函數(shù)SKIPIF1<0則滿足SKIPIF1<0的SKIPIF1<0取值范圍是A．[-1,2] B．[0,2] C．[1,+SKIPIF1<0） D．[0,+SKIPIF1<0）3．若關(guān)于x的不等式SKIPIF1<0有實(shí)數(shù)解，則實(shí)數(shù)a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．若不等式SKIPIF1<0恒成立，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題5．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0,則實(shí)數(shù)SKIPIF1<0的取值范圍為___________.6．已知函數(shù)SKIPIF1<0的圖象關(guān)于原點(diǎn)對稱，若SKIPIF1<0，則SKIPIF1<0的取值范圍為________．三、解答題7．解下列方程：(1)SKIPIF1<0；(2)SKIPIF1<0；(3)SKIPIF1<0；(4)SKIPIF1<0．8．已知函數(shù)SKIPIF1<0，與SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對稱的圖象過點(diǎn)SKIPIF1<0.(1)求SKIPIF1<0的值；(2)求不等式SKIPIF1<0的解集.題型四指數(shù)函數(shù)的綜合應(yīng)用策略方法指數(shù)函數(shù)通過平移、伸縮及翻折等變換，或與其他函數(shù)進(jìn)行結(jié)合形成復(fù)合函數(shù)時，我們對這類問題的解決方式是進(jìn)行還原分離，化繁為簡，借助函數(shù)的單調(diào)性、奇偶性、對稱性及周期性解決問題．【典例1】函數(shù)SKIPIF1<0單調(diào)遞增區(qū)間為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】當(dāng)SKIPIF1<0時，不等式SKIPIF1<0恒成立，則實(shí)數(shù)a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】已知SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，對任意正數(shù)SKIPIF1<0，SKIPIF1<0，都有SKIPIF1<0，且SKIPIF1<0，當(dāng)SKIPIF1<0時，SKIPIF1<0，則不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．函數(shù)SKIPIF1<0，下列關(guān)于函數(shù)SKIPIF1<0的說法錯誤的是（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于原點(diǎn)對稱B．函數(shù)SKIPIF1<0的值域?yàn)镾KIPIF1<0C．不等式SKIPIF1<0的解集是SKIPIF1<0D．SKIPIF1<0是增函數(shù)2．已知函數(shù)SKIPIF1<0，使不等式SKIPIF1<0成立的一個必要不充分條件是（

）A．SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0 C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<03．已知SKIPIF1<0為偶函數(shù)，SKIPIF1<0為奇函數(shù)，且滿足SKIPIF1<0.若對任意