2. 外语
  3. The function f has the property that f(x) = f(x+1) for all numbers x. And g(x) = f(x)+1. Which of the following statements must

The function f has the property that f(x) = f(x+1) for all numbers x. And g(x) = f(x)+1. Which of the following statements must

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问题 The function f has the property that f(x) = f(x+1) for all numbers x.
And g(x) = f(x)+1. Which of the following statements must be true?
(Ⅰ) In a rectangular coordinate, g(x) is a horizontal line parallel to x axis.
(Ⅱ) In a rectangular coordinate, g(x) is a horizontal line parallel to y axis.
(Ⅲ) If g(1) = 9, f(4) = 8.
选项 A、(Ⅰ)(Ⅲ)
B、(Ⅱ)(Ⅲ)
C、(Ⅲ) only
D、None of the above.
答案C
解析 根据题目f(x)=f(x+1)得知，f(x)为周期函数。g(x)=f(x)+1表示g(x)的图像是f(x)的图像向上移动一个单位后得到的。周期函数并非一定是直线，即f(x)和g(x)都不一定是直线，也就不一定平行于x轴或y轴，所以(Ⅰ)、(Ⅱ)不能选。或使用列举法判断：假设f(0)=f(1)可以取值1，f(0.5)=f(1.5)可以取值0.5，此时f(x)的图像不一定是直线，(Ⅰ)、(Ⅱ)不能选。g(1)=9→f(1)+1=9→f(1)=8→f(4)=8，所以(Ⅲ)是正确的，答案为C。
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