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单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,2016-03-13,#,指数与指数函数,一、整数指数幂的运算性质,二、根式的概念,如果一个数的,n,次方等于,a,(,n,1,且,n,N,*,),那么这个数叫,做,a,的,n,次方根,.,即,:,若,x,n,=,a,则,x,叫做,a,的,n,次方根,其中,n,1,且,n,N,*,.,式子,a,叫做根式,这里,n,叫做,根指数,a,叫做,被开方数,.,n,(1),a,m,a,n,=,a,m+n,(,m,n,Z);,(2),a,m,a,n,=,a,m,-,n,(,a,0,m,n,Z);,(3)(,a,m,),n,=,a,mn,(,m,n,Z);,(4)(,ab,),n,=,a,n,b,n,(,n,Z).,三、根式的,性质,5.,负数没有偶次方根,.,6.,零的任何次方根都是零,.,1.,当,n,为奇数时,正数的,n,次方根是一个正数,负数的,n,次方根是一个负数,a,的,n,次方根用符号,a,表示,.,n,2.,当,n,为偶数时,正数的,n,次方根有两个,它们互为相反数,这时,正数的正的,n,次方根用符号,a,表示,负的,n,次方根用符号,-,a,表示,.,正负两个,n,次方根可以合写为,a,(,a,0).,n,n,n,3.(,a,),n,=,a,.,n,4.,当,n,为奇数时,a,n,=,a,;,n,当,n,为偶数时,a,n,=|,a,|=,n,a,(,a,0),-,a,(,a,0,且,a,1),叫做,指数函数,其中,x,是自变量,函数的定义域是,R.,六、指数函数,a,=,a,m,a,-,=,(,a,0,m,n,N,*,且,n,1).,n,m,n,n,m,n,m,a,1,(1),a,r,a,s,=,a,r+s,(,a,0,r,s,Q,);,(2),a,r,a,s,=,a,r,-,s,(,a,0,r,s,Q,);,(3)(,a,r,),s,=,a,rs,(,a,0,r,s,Q,);,(4)(,ab,),r,=,a,r,b,r,(,a,0,b,0,r,Q,).,图,象,性,质,y,o,x,(0, 1),y,=1,y,=,a,x,(,a,1),a,1,y,o,x,(0, 1),y,=1,y,=,a,x,(0,a,1),0,a,0,a,1),图象经过第二、三、四象限,则一定有,( ),A. 0,a,0 B.,a,1,b,0 C. 0,a,1,b,1,b,0,2.,若,0,a,1,b,a,b,B.,b,a,c,C.,a,b,c,D.,a,c,b,1,2,4.,若,0,a,b,(1,-,a,),b,B. (1+,a,),a,(1+,b,),b,C. (1,-,a,),b,(1,-,a,) D. (1,-,a,),a,(1,-,b,),b,b,1,2,b,C,A,D,D,C,5.,设,a,=6,0.7,b,=0.7,6,c,=log,0.7,6,则,( ),A.,c,a,b,B.,b,a,c,C.,a,b,c,D.,a,c,b,典型例题,1.,化简下列各式,:,(1) (1,-,a,) ;,(,a,-,1),3,1,4,(2),xy,2,xy,-,1,xy,;,3,4,=,-,a,-,1,.,=,xy,.,解,:,(1),原式,=,(1,-,a,)(,a,-,1),-,4,3,=,-,(,a,-,1)(,a,-,1),-,4,3,=,-,(,a,-,1),4,1,(2),原式,=,xy,2,(,xy,-,1,),(,xy,),2,1,3,1,2,1,=(,xy,2,x,y,-,),x,y,3,1,2,1,2,1,2,1,2,1,=(,x y,),x,y,2,3,2,3,3,1,2,1,2,1,=,x,y,x,y,2,1,2,1,2,1,2,1,(3) (1,-,a,)(,a,-,1),-,2,(,-,a,) .,2,1,2,1,a,-,11),求,的值,.,a,1,x,-,x,2,-,1,x,2,-,1,解,:,以,x,+,x,2,-,1,、,x,-,x,2,-,1,为根构造方程,:,t,2,-,2,xt,+1=0,即,:,t,2,-,(,a,+ ),t,+,a,=0,a,1,a,1,a,1,t,=,a,或,.,x,+,x,2,-,1 ,x,-,x,2,-,1,a,1,x,-,x,2,-,1 = .,x,+,x,2,-,1 =,a,a,1,x,2,-,1 = (,a,-,),1,2,a,1,原式,=,(,a,-,),1,2,a,1,a,1,= (,a,-,1).,1,2,解法二,:,将已知式整理得,:,(,a,),2,-,2,x,a,+1=0,或,(,),2,-,2,x,(,)+1=0.,a,1,a,1,a, ,a,1,a,=,x,+,x,2,-,1, =,x,-,x,2,-,1,a,1,以下同上,.,6.,已知函数,f,(,x,)=3,x,且,f,-,1,(18)=,a,+2,g,(,x,)=3,ax,-,4,x,的定义域为,0, 1. (1),求,g,(,x,),的解析式,; (2),求,g,(,x,),的单调区间,确定其增减性并用定义证明,; (3),求,g,(,x,),的值域,.,f,(,a,+2)=3,a,+2,=18.,解,:,(1),f,(,x,)=3,x,且,f,-,1,(18)=,a,+2,3,a,=2.,g,(,x,)=(3,a,),x,-,4,x,=2,x,-,4,x,.,即,g,(,x,)=2,x,-,4,x,.,(2),令,t,=2,x,则,函数,g,(,x,),由,y,=,t,-,t,2,及,t,=2,x,复合而得,.,由已知,x,0, 1,则,t,1, 2,t,=2,x,在,0, 1,上单调递增,y,=,t,-,t,2,在,1, 2,上单调递减,g,(,x,),在,0, 1,上单调递减,证明如下,:,g,(,x,),的定义域区间,0, 1,为函数的单调递减区间,.,对于任意的,x,1,x,2,0, 1,且,x,1,x,2,g,(,x,1,),-,g,(,x,2,),0,x,1,x,2,1,2,x,1,-,2,x,2,0,且,1,-,2,x,1,-,2,x,2,g,(,x,2,).,故函数,g,(,x,),在,0, 1,上单调递减,.,=(2,x,1,-,4,x,1,),-,(2,x,2,-,4,x,2,),=(2,x,1,-,2,x,2,),-,(2,x,1,-,2,x,2,)(2,x,1,+2,x,2,),=(2,x,1,-,2,x,2,)(1,-,2,x,1,-,2,x,2,),=(2,x,1,-,2,x,2,)(1,-,2,x,1,-,2,x,2,)0.,x,0, 1,时有,:,解,:,(3),g,(,x,),在,0, 1,上单调递减,g,(1),g,(,x,),g,(0).,g,(1)=2,1,-,4,1,=,-,2,g,(0)=2,0,-,4,0,=0,-,2,g,(,x,),0,.,故,函数,g,(,x,),的值域为,-,2, 0.,6.,已知函数,f,(,x,)=3,x,且,f,-,1,(18)=,a,+2,g,(,x,)=3,ax,-,4,x,的定义域为,0, 1. (1),求,g,(,x,),的解析式,; (2),求,g,(,x,),的单调区间,确定其增减性并用定义证明,; (3),求,g,(,x,),的值域,.,绝世唐门,http:/www.adf.cc/15707/,岨銵琚,7.,设,a,0,f,(,x,)=,-,是,R,上的奇函数,. (1),求,a,的值,; (2),试判断,f,(,x,),的反函数,f,-,1,(,x,),的奇偶性与单调性,.,a,e,x,a,e,x,解,:,(1),f,(,x,),是,R,上的奇函数,f,(0)=0,即,-,a,=0.,1,a,a,2,=1.,a,0,a,=1.,(2),由,(1),知,f,(,x,)=,e,x,-,e,-,x,x,R,f,(,x,),R.,f,(,x,),是奇函数,f,(,x,),的反函数,f,-,1,(,x,),也是奇函数,.,y,=,e,-,x,是,R,上的减函数,y,=,-,e,-,x,是,R,上的增函数,.,又,y,=,e,x,是,R,上的增函数,y,=,e,x,-,e,-,x,是,R,上的增函数,.,f,(,x,),的反函数,f,-,1,(,x,),也是,R,上的增函数,.,综上所述,f,-,1,(,x,),是奇函数,且是,R,上的增函数,.,此时,f,(,x,)=,e,x,-,e,-,x,是,R,上的奇函数,.,a,=1,即为所求,.,
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