已知函数f（x）的定义域为R，对任意s，t∈R都有f（s+t）=f（s）+f（t_爱下问搜题

搜题请搜索小程序“爱下问搜题”

问题解答题

已知函数f（x）的定义域为R，对任意s，t∈R都有f（s+t）=f（s）+f（t），且对任意x＞0，都有f（x）＜0，且已知f（3）=-3．

（1）求证：f（x）是R上的单调递减函数；

（2）求证：f（x）是奇函数；

（3）求f（x）在[m，n]（m，n∈Z且m＞0）上的值域．

答案

（1）在R任取x₁，x₂，且x₁＜x₂，则f（x₂）=f[（x₂-x₁）+x₁]=f（x₂-x₁）+f（x₁），…（1分）

∴f（x₂）-f（x₁）=f（x₂-x₁）．…（2分）

∵x₂-x₁＞0，∴f（x₂-x₁）＜0，…（3分）∴f（x₂）-f（x₁）＜0，即f（x₂）＜f（x₁），∴f（x）是R上的单调递减函数． …（4分）

（2）令s=t=0，则f（0）=f（0）+f（0），∴f（0）=0．…（5分）

又令s=x，t=-x，则f（0）=f（x）+f（-x），∴f（x）+f（-x）=0，…（6分）

∴f（-x）=-f（x），∴f（x）是奇函数．…（7分）

（3）∵f（x）是R上的单调递减函数，∴f（x）在[m，n]上也为减函数，…（8分）

∴f（x）在[m，n]上的最大值为f（m），最小值为f（n）．…（9分）

又m，n∈Z，∴f（m）=f[1+（m-1）]=f（1）+f（m-1）=2f（1）+f（m-2）=…=mf（1）．

同理f（n）=nf（1），…（11分）

已知f（3）=-3得f（3）=3f（1）=-3，∴f（1）=-1，…（12分）∴f（n）=-n，f（m）=-m，…（13分）

所以，函数的值域为[-n，-m]．…（14分）

单项选择题 A1/A2型题

能引起肾小球肾炎的是（）

A.链球菌

B.变形杆菌

C.李斯特菌

D.布鲁菌

E.铜绿假单胞菌

完形填空

“Sixteen,” I said. I have forgotten the math question my second-grade teacher, Joyce Cooper, asked that day, but I will never forget my   36  . As soon as the number left my mouth, the whole class started 37  . I felt like the   38  person in the world..
Mrs. Cooper fixed them with a firm look. Then she said: “We’re all here to 39 .”
Once Mrs. Cooper asked us to write a 40 about what we hoped to do in the future. I wrote: “I want to be a 41 like Mrs. Cooper.
She wrote on my report, “You would 42 an outstanding teacher 43  you are determined and you try hard.” I was to carry those 44 in my heart for the 27 years.
After I graduated from high school, I got married and had children.
We needed every penny just to 45 . College and teaching were out of the question.
Then one day I thought of my 46 of how I wanted to help children. I talked it over with my family and 47  to attend college classes in the 48 before work. And when I got home from work, I would 49 . Finally, after seven years, my dream had been realized and I became a teacher.
Not long after I started 50 , something happened that brought the 51 rushing back to me. I had written a sentence with grammatical errors on the blackboard. Then I asked students to come and 52 the mistakes.
One girl got halfway through, became 53 and stopped. As the other children laughed, 54 rolled down her cheeks. I gave her a hug and told her to have a drink of water. Then, remembering Mrs. Cooper, I fixed 55 of the class with a firm look. “We are all here to learn,” I said.

C．least amusing

D．most important