问题 解答题
已知f(x)=
1
x
-1

(1)求函数f(x)的定义域;
(2)判断并用定义证明函数f(x)的单调性;
(3)求函数f(x)的反函数f-1(x);
(4)若对任意满足x1+x2=m的正实数x1、x2,不等式f-1(x1)f-1(x2)>f-1(m)恒成立.求m的取值范围.
答案

(1)由

1
x
-1≥0得定义域为(0,1].

(2)f(x)在(0,1)内单调递减,证明如下.

设0<x1<x2≤1,则f(x2)-f(x1)=

1
x2
-1
-
1
x1
-1
=
x1-x2
x2x1
1
x2
-1
+
1
x1
-1
<0.

即f(x2)<f(x1).这就是说函数f(x)在(0,1]上单调递减.

(3)令y=

1
x
-1
,解得x=
1
1+y2
(y≥0),即f-1(x)=
1
1+x2
(x≥0).

(4)由f-1(x1)f-1(x2)>f-1(m),

化简得到:(1+x12)(1+x22)<1+m2

注意到m=x1+x2,以及x1,x2>0代入整理得:x1x2<2.

把x2=m-x1代入整理得到:x12-mx1+2>0.

该关于x1的不等式对于一切(0,m)内的x1恒成立.

所以(

m
2
)2-m•
m
2
+2>0.解得0<m<2
2

阅读理解

阅读下面短文并回答问题,然后将答案写到答题卷相应的位置上(请注意问题后的字数要求)。

Swimming is one of those activities that can be learned early in life. Little children can learn to swim as soon as they walk. In fact, you need the same skills in walking as in swimming. However, I believe that five is the best age to learn. By five or six, a child knows fear of water, a very important thing to know. It’s wise to be afraid, to recognize true danger. Young ones understand that the water can sometimes be very dangerous.

To really benefit from swimming, every swimmer should learn ____________:butterfly, backstroke, breaststroke, and crawl. I feel that one of these—the breaststroke—is different from the others, since some young swimmers use this stroke naturally, without any training.

In swimming, there are certain rules every swimmer should follow:

(1) Never swim alone! No matter how good you are in the water, don’t risk drowning by swimming alone. If you swim by yourself, with no life guards or friends with you, you may get into trouble.

(2) Don’t go beyond your abilities. Most swimmers know enough not to swim too far from the bank or the beach. Showing off by doing dangerous tricks is no good. Swim safely and you will continue to swim and alive.

(3) Don’t smoke. Swimming depends on a healthy body; good lungs are part of it.

(4) Work at any activity that builds muscles.

小题1:What is the main idea of the passage? (no more than 10 words)

___________________________________________________________ 

小题2:Please complete the following statement with proper words. (no more than 5 words)

__________________________and the recognition of its danger are sensible.

小题3:Fill in the blank in Paragraph 2 with proper words. (no more than 7 words)

_______________________________________________________________

小题4:Why should not you swim alone according to the passage? ( no more than 10 words)

_________________________________________________________________

小题5:What does the underlined word “it” refer to? (no more than 3 words)

________________________________________________________________

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