题目内容
(本小题满分13分)
已知函数
,设函数
,
(1)若
,且函数
的值域为
,求
的表达式.
(2)若
在
上是单调函数,求实数
的取值范围.
已知函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232335247001732.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524716887.png)
(1)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524747470.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524778429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524794447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524778429.png)
(2)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524840442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524856429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524872306.png)
(1) 由![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524903657.png)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524918907.png)
当
时,
,
在
上单调,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525277195.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525293466.png)
当
时,
① 当
时,
或![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525574645.png)
②当
时,
或
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524903657.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524918907.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524950235.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524965359.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525074761.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525090482.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524856429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525277195.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525293466.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525449291.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525496375.png)
① 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525542381.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525558661.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525574645.png)
②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525605381.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525620644.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525652661.png)
本试题主要是考查了二次函数的性质和二次函数的解析式的综合运用。
(1)
的值域为
,同时函数在x=1处的函数值为零,得到参数a,b的值。
(2)根据函数在给定区间是单调函数, 需要对于函数的性质和对称轴的位置分情况讨论得到。
(1)显然
分
的值域为
分
由
(7分)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524918907.png)
当
时,
,
在
上单调,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525277195.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525293466.png)
当
时,
图象满足:对称轴:
在
上单调
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525277195.png)
或
……………………11分
② 当
时,
或![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525574645.png)
②当
时,
或
综上:略----13分
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525683641.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525698812.png)
(2)根据函数在给定区间是单调函数, 需要对于函数的性质和对称轴的位置分情况讨论得到。
(1)显然
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525496375.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525854801.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525683641.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525901927.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232335260421856.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524918907.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524950235.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524965359.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525074761.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525090482.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524856429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525277195.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525293466.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525449291.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525496375.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233526416426.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233526588669.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525090482.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233524856429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525277195.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233526666723.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233526697716.png)
② 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525542381.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525558661.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525574645.png)
②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525605381.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525620644.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233525652661.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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