问题 解答题
设函数f(x)的定义域关于原点对称,对定义域内任意的x存在x1和x2,使x=x1-x2,且满足:
(1)f(x1-x2)=
f(x1)-f(x2)
1+f(x1)•f(x2)

(2)当0<x<4时,f(x)>0
请回答你列问题:
(1)判断函数的奇偶性并给出理由;
(2)判断f(x)在(0,4)上的单调性并给出理由.
答案

(1)函数f(x)在定义域内是奇函数.

因为在定义域内,对任意x存在x1和x2,使x=x1-x2,且满足:f(x1-x2)=

f(x1)-f(x2)
1+f(x1)•f(x2)

由于函数f(x)的定义域关于原点对称,-x必与x同时在定义域内,

同样存在x1和x2,使-x=x2-x1,且满足:f(-x)=f(x2-x1)=

f(x2)-f(x1)
1+f(x2)•f(x1)
,即f(x)=-f(-x),

∴f(-x)=-f(x),

∴函数f(x)在定义域内是奇函数.

(2)函数f(x)在(0,l)上是单调递增函数.

任意取x1,x2∈(0,l),且x1<x2,则x2-x1>0,

∵函数f(x)在定义域内是奇函数,且当0<x<l时,f(x)>0,

∴f(x1)>0,f(x2)>0,f(x1-x2)=-f(x2-x1)<0,

又∵f(x1-x2)=

f(x1)-f(x2)
1+f(x1)•f(x2)

∴f(x1)-f(x2)<0,∴f(x1)<f(x2),

∴函数f(x)在(0,l)上是单调递增函数.

阅读理解

阅读理解。

     Games are fun. After hours of homework it's great to relax by playing a game. But do games ever do

kids any harm (害处)? And are there any ways parents can help kids stay away from bad games? The US

researchers are trying to find solutions to these problems.

     They studied what video games could do to children. The researchers found game sellers had made big

changes to make sure the games didn't harm kids. And they suggested that work should be done to find out

if video games affected children's school work and health.

     One problem found was that parents hadn't taken enough notice of their kids' game-playing habits.

     "Many game sellers put control (控制键) on the games to help parents make sure that children don't play

games unsuitable for them," said David Walsh, the president (院长) of the National Institute on Media and

the Family. Parents should watch the games kids are playing and know how much time they are spending

playing them.

     The study found that playing games too long could affect kids' health as well as their school work. Almost

half of all "heavy garners" were six to seventeen years old. These kids were more likely to have weight and

eyesight problems than other kids. Playing too many games also affected how children did at school.

     Kids are supposed to ask their parents to set a time limit (限制) for playing games and remind them

when the time is up.

1. The US researchers are trying to find out _____.

A. whether games do any good to kids

B. if there are ways for parents to keep kids away from bad games

C. how many students will spend their most time playing games

D. whether games are fun

2. The word "affected"in the second paragraph probably means _____.

A. continued

B. improved

C. had influences on

D. were divided between

3. What problems did the researchers find out about parents?

A. Parents were too strict with their kids.

B. Parents cared too much about kids' study skills.

C. Parents didn't notice their kids' hobbies.

D. Parents didn't pay much attention to their kids' game.playing habits.

4. According to the passage,playing too many games can affect a kid's _____.

A. weight and height

B. health and school work

C. brain and thought

D. manners

5. The main purpose of this passage is that _____.

A. kids are supposed to have a time limit for playing games with their parents' help

B. parents must join their kids in playing games

C. parents should learn to play games from their children

D. kids can play games with the help of their parents

多项选择题