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2017上海中学高一期末

上海中学2016学年第一学期期末考试数学试卷

2017.1

一. 填空题

函数2

()lg(31)f x x =++的定义域为 2. 函数2()f x x =（1x ≥）的反函数为1()f x -=

3. 若幂函数()f x 的图像经过点1(27,)9，则该函数解析式为()f x =

4. 若对任意不等于1的正数a ，函数2()3x f x a

+=-的图像都过点P ，则点P 的坐标是

5. 已知2

()f x ax bx =+是定义在[3,2]a a -上的偶函数，那么a = ，b =

6. 方程224log (1)log (1)5x x +++=的解为

7. 已知符号函数10sgn()0010x x x x >??==??-

，则函数sgn(||)|sgn()|y x x =+的值域为

8. 已知()f x 是定义在R 上的奇函数，当0x <时，2()f x x x =+，则函数()f x 的解析式为()f x =

9. 函数2|65|0.3x x y -+=的单调增区间为

10. 设函数()y f x =存在反函数1()f x -，若满足1()()f x f x -=恒成立，则称()f x 为“自

反函数”，如函数()f x x =，()g x b x =-，()k h x x

=（0k ≠）等都是“自反函数”，试写出一个不同于上述例子的“自反函数”y =

11. 方程2

210x x +-=的解可视为函数2y x =+的图像与函数1y x

=的图像交点的横坐标，若方程440x ax +-=的各个实根12,,,k x x x ???（4k ≤）所对应的点4(,)i i

x x （1,2,,i k =???）均在直线y x =的同侧，则实数a 的取值范围是 12. 对于函数()y f x =，若存在定义域D 内某个区间[,]a b ，使得()y f x =在[,]a b 上的值域也是[,]a b ，则称函数()y f x =在定义域D 上封闭，如果函数()1||

kx f x x =

+（0k ≠）在R 上封闭，那么实数k 的取值范围是

二. 选择题

13. 已知3

()1f x ax bx =++（0ab ≠），若(2017)f k =，则(2017)f -=（）

A. k

B. k -

C. 1k -

D. 2k -

14. 定义在R 上的函数()y f x =在区间(,2)-∞上是增函数，且函数(2)y f x =+的图像关于直线1x =对称，则（）

A. (1)(5)f f <

B. (1)(5)f f >

C. (1)(5)f f =

D. (0)(5)f f =

15. 汽车的“燃油效率”是指汽车每消耗1升汽油行使的里程，下图描述了甲、乙、丙三辆汽车在不同速度下得燃油效率情况，下列叙述中正确的是（）

A. 消耗1升汽油，乙车最多可行使5千米

B. 以相同速度行使相同路程，三辆车中，甲车消耗汽油最多

C. 甲车以80千米/小时的速度行使1小时，消耗10升汽油

D. 某城市机动车最高限速80千米/小时，相同条件下，在该市用丙车比用乙车更省油

16. 设函数2

|2|0()|log |0x x f x x x +≤?=?>?，若关于x 的方程()f x a =有四个不同的解1x 、2x 、 3x 、4x 且1234x x x x <<<，则312234

1()x x x x x ++的取值范围是（） A. [3,3)- B. (3,3]- C. (,3)-∞ D. (3,)-+∞

三. 解答题

17. 在平面直角坐标系中，作出下列函数的图像；

（1）1

3y x =；（2）||1

()12

x y =-；

18. 已知集合226{|3

10330,}x x D x x R +=-?+≤∈

，求函数2()log 2x f x =? （x D ∈）的值域；

19. 设函数()x x f x k a a

-=?-（0a >且1a ≠）是奇函数；（1）求常数k 的值；

（2）若8(1)3f =

，且函数22()2()x x g x a a mf x -=+-在区间[1,)+∞上的最小值为2-，求实数m 的值；

20. 已知函数()||1m f x x x

=+-；（1）当2m =时，判断()f x 在(,0)-∞上的单调性并证明；

（2）若对任意x R ∈，不等式(2)0x f >恒成立，求m 的取值范围；

（3）讨论函数()y f x =的零点个数；

21. 已知a R ∈，函数2()log [(3)34]f x a x a =-+-；

（1）当2a =时，解不等式1()0f x <；

（2）若函数2(4)y f x x =-的值域为R ，求a 的取值范围；

（3）若关于x 的方程21()log (2)0f x a x

-+=解集中恰好只有一个元素，求a 的取值范围；

参考答案

一. 填空题 1. 1(,1)3-

2. (1)x ≥

3. 23

x - 4. (2,2)-- 5. 1，0 6. 3x = 7. {0,2} 8. 22,0,0

x x x x x x ?-+≥??+

10. y (0)x ≥ 11. (,6)

(6,)-∞-+∞ 12. (,1)(1,)-∞-+∞

二. 选择题 13. D 14. C 15. D 16. B

三. 解答题

17. 略； 18. 1[,0]4

-； 19.（1）1k =；（2）2m =；

20.（1）递减；（2）14

m >

；（3）当11(,)(,)44m ∈-∞-+∞，1个零点；当11{,0,}44m ∈-，2个零点；当11(,0)(0,)44m ∈-，3个零点； 21.（1）1(,1)2

；（2）8a ≥；（3）2a =或3a =；

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