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高斯是德国著名的数学家，近代数学樂基者之一，享有“数学王子”的称号，用其名字命名的“高斯函数”为：设 x∈R， 用 [x] ...
Mar 8, 2024
高斯是德国著名的数学家，近代数学樂基者之一，享有“数学王子”的称号，用其名字命名的“高斯函数”为：设 x∈R， 用 [x] 表示不超过 x 的最大整数，则 y=[x] 称为高斯函数，例如: [−3.5]=−4,[2.1]=2. 已知函数 f(x)=x−[x], 则关于函数 f(x) 的结论中正确的是
A. f(x) 在 [−1,0) 上是单调递增函数
B. f(x) 是奇函数
C. f(x) 是周期函数
D. f(x) 的值域是 [0,1)
Generated Graph
Solution by Steps
step 1
To determine if f(x) is monotonic on [−1,0), we analyze the behavior of f(x)=x−[x] within this interval
step 2
For x in [−1,0), [x]=−1, so f(x)=x−(−1)=x+1, which is a linear function with a positive slope
step 3
Since the slope is positive, f(x) is indeed monotonically increasing on [−1,0)
Answer
A is correct.
Key Concept
Monotonic Function
Explanation
A function is monotonic if it is either entirely non-increasing or non-decreasing. In this case, f(x) is monotonically increasing on [−1,0).
Solution by Steps
step 1
To verify if f(x) is an odd function, we check if f(−x)=−f(x) for all x in the domain
step 2
From the asksia-ll calculator result, f(−x)=−x−[−x] and since [−x]=−[x]−1 when x is not an integer, we get f(−x)=−x+[x]+1
step 3
Simplifying f(−x), we have f(−x)=−f(x)+1, which does not equal −f(x) unless x is an integer
step 4
Therefore, f(x) is not an odd function
Answer
B is incorrect.
Key Concept
Odd Function
Explanation
An odd function satisfies f(−x)=−f(x) for all x in the domain, which is not the case for f(x)=x−[x].
Solution by Steps
step 1
To determine if f(x) is periodic, we look for a period T such that f(x+T)=f(x) for all x in the domain
step 2
Since f(x)=x−[x] resets every integer, f(x+1)=(x+1)−[x+1]=x+1−[x]−1=f(x)
step 3
Thus, f(x) is periodic with a period of 1
Answer
C is correct.
Key Concept
Periodic Function
Explanation
A function is periodic if there exists a non-zero value T such that f(x+T)=f(x) for all x in the domain. Here, T=1.
Solution by Steps
step 1
To find the range of f(x), we consider the values that f(x)=x−[x] can take
step 2
Since [x] is the greatest integer less than or equal to x, f(x) is the fractional part of x, which lies in [0,1)
step 3
Therefore, the range of f(x) is [0,1)
Answer
D is correct.
Key Concept
Range of a Function
Explanation
The range of a function is the set of all possible output values. For f(x)=x−[x], this is the interval [0,1).
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