指数函数学习

tonyfang posted @ 2015年8月21日 21:34 in math with tags math , 627 阅读

形如$y=a^x(a>0且a≠1)$的函数为指数函数,定义域为$R$。

关于$a$的限制的说明:

1.若$a=0$,那么当$x>0$时,$y$恒等于0;当$x\leq0$时,没有意义。

2.若$a<0$,则当$x$为一些分数时,$y$不在实数范围内,比如$a=-4,x=\frac{1}{2}$。

3.若$a=1$,$y=1^x=1$是个常函数,没有研究必要。

综上,$a>0且a≠1$。

性质:

1.定义域$R$,值域$(0,+\infty)$

2.$a^0=1$,所以过定点$(0,1)$

3.$0<a<1$时,为单调减函数,$a>1$时,为单调增函数(均在定义域范围内)。【这点与对数函数相同,可以联合记忆】。

4.$a^1=a$,函数过定点$(1,a)$,即当$x=1$时,$y$等于底数$a$。

5.$0<a<1$时,当$x<0$时,$y>1$;当$x>0$,$0<y<1$。

$a>1$时,当$x>0$时,$y>1$;当$x<0$,$0<y<1$。

6.非奇非偶函数,$y=a^x$与对数函数$y=log_{a}x$关于$y=x$对称。

简单的解释:

(1)$a$不确定时应分类讨论。

(2)$0<a<1,x→+\infty,y→0$,$a$的值越小,图像越靠近$y$轴,递减速度越快

           $a>1,x→-\infty,y→0$,$a$的值越大, 图像越靠近$y$轴,递增速度越快

(3)$y=a^x$与$y=(\frac{1}{a})^x$关于$y$轴对称。

比较函数大小:$0<x<+\infty$时,底大幂大;$-\infty<x<0$时,底小幂小。

 

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