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高一数学集合与函数检测试题与详细答案

高一数学集合与函数检测试题与答案 (满分 120 分) 第一卷
一、选择题 1 2 3 4 5 6 7 8 9 10

A. 2 x + 1 7.函数 y =

B. 2 x ? 1

C. 2 x ? 3 )

D. 2 x + 7

x x

+ x 的图象是(

5 分选、填题(75 分)

1.下列表示图形中的阴影部分的是( A. ( A U C ) I ( B U C ) B. ( A U B ) I ( A U C ) C. ( A U B ) I ( B U C )

) A B 8.若偶函数 f ( x ) 在 (? ∞,?1] 上是增函数,则下列关系式中成立的是( A. f ( ? ) < f ( ?1) < f ( 2) C. f ( 2) < f ( ?1) < f ( ? ) ) )

C D. ( A U B ) I C

3 2

3 2

3 2 3 D. f ( 2) < f ( ? ) < f ( ?1) 2
B. f ( ?1) < f ( ? ) < f ( 2)

2.若集合 A = {?1,1} , B = {x | mx = 1} ,且 A ∪ B = A ,则 m 的值为( A. 1 B. ?1 C. 1 或 ?1 D. 1 或 ?1 或 0

9.如果奇函数 f (x ) 在区间 [3, 7] 上是增函数且最大值为 5 , 那么 f (x ) 在区间 [? 7,?3] 上是( ) A.增函数且最小值是 ? 5 C.减函数且最大值是 ? 5 ) B.增函数且最大值是 ? 5 D.减函数且最小值是 ? 5

3.若集合 M = ( x, y ) x + y = 0 , N = ( x, y ) x + y = 0, x ∈ R, y ∈ R ,则有(
2 2

{

}

{

}

A. M U N = M

B. M U N = N

C. M I N = M )

D. M I N = ?

4.判断下列各组中的两个函数是同一函数的为( ⑴ y1 = ⑵ y1 =

( x + 3)( x ? 5) , y2 = x ? 5 ; x+3
x + 1 x ? 1 , y 2 = ( x + 1)( x ? 1) ;

10.下列四个命题:(1)函数 f ( x ) 在 x > 0 时是增函数, x < 0 也是增函数,所以 f (x ) 是增函数; (2)若函数 f ( x ) = ax 2 + bx + 2 与 x 轴没有交点,则 b ? 8a < 0 且 a > 0 ;(3) y = x ? 2 x ? 3
2
2

的递增区间为 [1, +∞ ) ;(4) y = 1 + x 和 y = 其中正确命题的个数是( A. 0 二、填空题 11.设全集 U = ( x, y ) x, y ∈ R ,集合 M = ?( x, y ) B. 1 ) D. 3

(1 + x) 2 表示相等函数。

⑶ f ( x) = x , g ( x) = ⑷ f ( x) =
3

x2 ;

x 4 ? x 3 , F ( x) = x 3 x ? 1 ;

C. 2

⑸ f1 ( x) = ( 2 x ? 5 ) 2 , f 2 ( x) = 2 x ? 5 。 A.⑴、⑵ B.⑵、⑶ C.⑷ D.⑶、⑸

{

}

? ?

y+2 ? = 1? , N = {( x, y ) y ≠ x ? 4} , x?2 ?

? x + 2( x ≤ ?1) ? 5.已知 f ( x) = ? x 2 ( ?1 < x < 2) ,若 f ( x) = 3 ,则 x 的值是( ?2 x( x ≥ 2) ?
A. 1 B. 1 或



那么 (CU M ) I (CU N ) 等于_____ ___________。 12.若函数 f ( x ) = 13.已知 f ( x ) = ?

3 2

C. 1 ,

3 或± 3 2

x+a 在 [ ?1,1] 上是奇函数,则 f ( x ) 的解析式为 x + bx + 1
2

D. 3 )

6.设函数 f ( x) = 2 x + 3, g ( x + 2) = f ( x) ,则 g ( x) 的表达式是(

?1, x ≥ 0 ,则不等式 x + ( x + 2) ? f ( x + 2) ≤ 5 的解集是 ?? 1, x < 0

14.函数 y =

(x + 1)0
x ?x

的定义域是____ _________________。

15.已知 A = y y = ? x 2 + 2 x ? 1 , B = y y = 2 x + 1 ,则 A I B = _________。

{

}

{

}

第二卷 综合应用题(15 分/道,45 分)
三、解答题: 16.设全集 U = R ,

M = {m | 方程mx 2 ? x ? 1 = 0有实数根} N = {n | 方程x 2 ? x + n = 0有实数根} , 求 ( CU M ) I N .



且当 x ∈ [ ?1, 2] 时, f ( x ) 18. 已知 g ( x ) = ? x ? 3 , f ( x ) 是二次函数,g ( x ) + f ( x ) 是奇函数,
2

的最小值是 1,求 f ( x ) 的表达式. 17. 已知函数 f ( x ) 的定义域为 ( ?1,1) ,且同时满足下列条件: (1) f ( x ) 是奇函数; (2) f ( x ) 在定义域上单调递减; (3) f (1 ? a ) + f (1 ? a 2 ) < 0, 求 a 的取值范围。

(2) 当 ?1 ≤ ?

b b b2 b2 ≤ 2 即 ?4 ≤ b ≤ 2 时, f ( x )min = f ? ? ? = ? + 3 = 1, ? ? 2 2 ? 2? 4

第一章单元检测题答案
一、 (每题 4 分,共 40 分)
1 A 2 D 3 A 4 C 5 D 6 B 7 D
2

解得 b = ?2 2 或 b = 2 2 (舍) ; (3) 当 ?
2

8 D

9 A

10 A

b , > 2 即 b < ?4 时, f ( x ) min = f ( 2 ) = 7 + 2b = 1,∴ b = ?3 (舍) 2

二、11、 {(2,?2)} 13、 ( ?∞, ]

12、 f ( x ) =

x x +1

2 综上知 f ( x ) = x + 3x + 3 或 f ( x ) = x ? 2 2 + 3 .

3 2

14、 (? ∞,?1) ∪ (? 1,0)

15、 { y | y ≤ 0}
三、 16.解:当 m = 0 时, x = ?1 ,即 0 ∈ M ; 当 m ≠ 0 时, ? = 1 + 4 m ≥ 0, 即 m ≥ ? ∴m ≥ ?

1 ,且 m ≠ 0 4

1 1? ? ,∴ CU M = ? m | m < ? ? 4 4? ?
1 1? ? ,∴ N = ? n | n ≤ ? 4 4? ?

而对于 N , ? = 1 ? 4n ≥ 0, 即 n ≤

∴ (CU M ) I N = ? x | x < ? ?

? ?

1? 4?

? ?1 < 1 ? a < 1 ? 17.解: f (1 ? a ) < ? f (1 ? a ) = f ( a ? 1) ,则 ? ?1 < 1 ? a 2 < 1 , ?1 ? a > a 2 ? 1 ?
2 2

∴ 0 < a <1
18.解:设 f ( x ) = ax + bx + c ( a ≠ 0 ) ,则
2

f ( x ) + g ( x ) = ( a ? 1) x 2 + bx + c ? 3, 又 f ( x ) + g ( x ) 为奇函数, ∴ ( a ? 1) x 2 ? bx + c ? 3 = ? ( a ? 1) x 2 ? bx ? c + 3 对 x ∈ R 恒成立,
?a ? 1 = ?a + 1 ?a = 1 ∴? ,解得 ? , ?c ? 3 = ? c + 3 ?c = 3
b ∴ f ( x ) = x 2 + bx + 3 ,其对称轴为 x = ? . 2 b (1) 当 ? < ?1 即 b ≥ 2 时, f ( x )min = f ( ?1) = 4 ? b = 1,∴ b = 3 ; 2


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