问题 解答题
已知函数f(x)=log
1
2
(x+1),当点P(x0,y0)在y=f(x)的图象上移动时,点Q(
x0-t+1
2
,y0)(t∈R)在函数y=g(x)的图象上移动.
(1)若点P坐标为(1,-1),点Q也在y=f(x)的图象上,求t的值;
(2)求函数y=g(x)的解析式;
(3)当t>0时,试探求一个函数h(x)使得f(x)+g(x)+h(x)在限定定义域为[0,1)时有最小值而没有最大值.
答案

(1)当点P坐标为(1,-1),点Q的坐标为(

1-t+1
2
,-1),

∵点Q也在y=f(x)的图象上,∴-1=log

1
2
(-1+
t
2
+1),即t=0.

(根据函数y=f(x)的单调性求得t=0,请相应给分)

(2)设Q(x,y)在y=g(x)的图象上

x=
x0-t+1
2
y=y0
,即
x0=2x+t-1
y0=y

而P(x0,y0)在y=f(x)的图象上,∴y0=log

1
2
(x0  +1)

代入得,y=g(x)=log

1
2
(2x+t)为所求.

(3)h(x)=log

1
2
1-x
2x+t
;或h(x)=log
1
2
3
2
-x
2x+t
等.

如:当h(x)=log

1
2
1-x
2x+t
时,

f(x)+g(x)+h(x)=log

1
2
 (x+1)+log
1
2
(2x+t)+log
1
2
1-x
2x+t
=log
1
2
 (1-x2)

∵1-x2在[0,1)单调递减,∴0<1-x2≤1故log

1
2
(1-x2)≥0,

即f(x)+g(x)+h(x)有最小值0,但没有最大值.

单项选择题
单项选择题

There is virtually no limit to how one can serve community interests, from spending a few hours a week with some charitable organization to practically full-time work for a social agency. Just as there are opportunities for voluntary service (1) (VSO) for young people before they take up full-time employment, (2) there are opportunities for overseas service for (3) technicians in developing countries. Some people, (4) those who retire early, (5) their technical and business skills in countries (6) there is a special need.

So in considering voluntary or (7) community service, there are more opportunities than there (8) were when one first began work. Most voluntary organizations have only a small full-time

(9) and depend very much on volunteers and part-timers. This means that working relationships are different from those in commercial organizations, and values may be different. (10) some ways they may seem more casual and less efficient, but one should not (11) them by commercial criteria. The people who work with them do so for different reasons and with different (12) , both personal and (13) . One should not join them (14) to arm them with professional experience; they must be joined with commitment to the (15) , not business efficiency. Because salaries are (16) or non-existent many voluntary bodies offer modest expense. But many retired people take part in community service for (17) , simply because they enjoy the work.

Many community activities possible (18) retirement were also possible during one’s working life but they are to be undertaken (19) seriously for that. Retired people who are just looking for something different or unusual to do should not consider (20) community service.

2()

A.as

B.so

C.then

D.that