问题 解答题
定义在R上的函数f(x)满足f(x+y)+1=f(x)+f(y)(x,y∈R),f(1)=0,且当x>1时f(x)<0.
(1)证明:f(x)在R上是减函数;
(2)若4f(
m+1
4
)≥3
,求实数m的范围.
答案

(1)证明:取y=1,则f(x+1)+1=f(x)+f(1)=f(x).设x1,x2∈R,且x1>x2,则x1-x2>0,x1-x2+1>1,

因为当x>1时f(x)<0,所以f(x1-x2+1)<0.

f(x1)=f(x1+1)+1=f[x2+(x1-x2+1)]+1

=f(x2)+f(x1-x2+1)-1+1=f(x2)+f(x1-x2+1).

因为f(x1-x2+1)<0,所以f(x2)<f(x1).

所以函数f(x)在R上是减函数;

(2)取x=y=0,得f(0)+1=f(0)+f(0),

所以f(0)=1,

4f(

m+1
4
)≥3,得4f(
m+1
4
)=4f(
m
4
)-1≥3

所以4f(

m
4
)≥4,f(
m
4
)≥1

因为f(x)为实数集上的减函数,且f(0)=1

所以

m
4
≤0.

则m≤0.

所以实数m的范围是(-∞,0].

阅读理解

May the first is an important date in the college admission process in the United States. This is the last day for high school seniors to accept or reject offers of admission in the fall. But according to a recent report, there is a great change. Acceptance rates at the top colleges this year were lower than ever. 小题1:       

These days, college applicants are applying to more colleges because online and common applications make the process easier. 小题2:         On the other hand, deciding students for top colleges to accept can be difficult because it takes much time. Students may not like being put on waiting lists. Also, common colleges work more to get the students they accept to accept them. Some colleges hold visiting days that offer a chance to attend classes and stay overnight. Some universities send gifts. 小题3:       .

For many families, the most important thing colleges can do is to show them the money, especially this year; the weak economy makes parents nervous. 小题4:       . The recent problems that spread from the housing market to credit markets have even affected student’s loans. So lately, top schools have to spare lots of money to increase their financial aid for students.

 小题5:        . The Education Department expects the number of high school graduates to start to go down. This will happen as the last of the children of the baby boom generation finishes high school. The number is not expected to start rising again until 2015.

A.Besides, top colleges are facing changes in the population.

B.What contributes to it?

C.But a mistake can be costly if it happens.

D.But top colleges ignore them.E. They have international students who know English.

F. It can mean several acceptances to choose from.

G. They cannot afford college as planned.

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