题目内容
对于函数
,有下列五个命题:
①若
存在反函数,且与反函数图象有公共点,则公共点一定在直线
上;
②若
在
上有定义,则
一定是偶函数;
③若
是偶函数,且
有解,则解的个数一定是偶数;
④若
是函数
的周期,则
,也是函数
的周期;
⑤
是函数
为奇函数的充分不必要条件。
从中任意抽取一个,恰好是真命题的概率为 ( )
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632290568.png)
①若
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632290568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632383408.png)
②若
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632290568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632430310.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632446614.png)
③若
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632290568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632555536.png)
④若
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632633547.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632290568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632680717.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632290568.png)
⑤
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632711501.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632290568.png)
从中任意抽取一个,恰好是真命题的概率为 ( )
A.![]() | B.![]() | C.![]() | D.![]() |
B
①若y=f(x)存在反函数,且与反函数图象有公共点,则公共点不一定在直线y=x上,如函数f(x)=
,反函数是其本身,公共点是整个函数图象;
②若y=f(x)在R上有定义,则y=f(|x|)一定是偶函数,因f(|-x|)=f(|x|)对于任意x恒成立,故正确;
③若y=f(x)是偶函数,且f(x)=0有解,则解的个数一定是偶数不正确,如y=x2,是偶函数,x2=0的解只有一个,不是偶数个;
④若T(T≠0)是函数y=f(x)的周期,则f(x+T)=f(x),从而f(x+nT)=f(x),则nT(n∈N),也是函数y=f(x)的周期;
⑤f(0)=0是函数y=f(x)为奇函数的充分也不必要条件,不正确,f(x)=x2时,f(0)=0,而f(x)=x2是偶函数.
故正确的命题有2个,
则从中任意抽取一个,恰好是真命题的概率为![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632773374.png)
故选B.
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632945315.png)
②若y=f(x)在R上有定义,则y=f(|x|)一定是偶函数,因f(|-x|)=f(|x|)对于任意x恒成立,故正确;
③若y=f(x)是偶函数,且f(x)=0有解,则解的个数一定是偶数不正确,如y=x2,是偶函数,x2=0的解只有一个,不是偶数个;
④若T(T≠0)是函数y=f(x)的周期,则f(x+T)=f(x),从而f(x+nT)=f(x),则nT(n∈N),也是函数y=f(x)的周期;
⑤f(0)=0是函数y=f(x)为奇函数的充分也不必要条件,不正确,f(x)=x2时,f(0)=0,而f(x)=x2是偶函数.
故正确的命题有2个,
则从中任意抽取一个,恰好是真命题的概率为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825135632773374.png)
故选B.
![](http://thumb2018.1010pic.com/images/loading.gif)
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