Sorry to post the same question again, but I haven't gotten any answers on the last time I asked which was roughly 9 hours ago. Here's the question:
We let g(x) = ⌊x⌊x⌋⌋ for x >= 0. Find the number of possible values of g(x) for 0 <= x <= 10.
Please help ASAP; I've been working on Algebra all day and this is the one problem that I still cannot solve.
Thanks.
https://www.jiskha.com/questions/1827797/hello-ive-been-having-trouble-on-the-following-question-we-let-g-x-x-x
You need to be ready to explain to a math tutor what you've tried and where you get stuck.
Ah, okay.
I apologize for the delay in receiving an answer to your question. I'm here to help you find the number of possible values of g(x) for 0 <= x <= 10.
To solve this problem, we first need to understand the notation used. The symbol ⌊x⌋ represents the floor function, which gives the largest integer less than or equal to x.
Now, let's determine the possible values of g(x) for 0 <= x <= 10 step by step:
1. When x = 0, g(x) = ⌊0⌊0⌋⌋ = ⌊0⋅0⌋ = ⌊0⌋ = 0.
2. When 0 < x < 1, g(x) = ⌊x⌊x⌋⌋ = ⌊x⋅0⌋ = ⌊0⌋ = 0. Since the floor function of any number between 0 and 1 inclusive is always 0, the value of g(x) does not change.
3. When x = 1, g(x) = ⌊1⌊1⌋⌋ = ⌊1⋅1⌋ = ⌊1⌋ = 1.
4. When 1 < x < 2, g(x) = ⌊x⌊x⌋⌋ = ⌊x⋅1⌋ = ⌊x⌋. Here, the floor function of x will always be an integer between 1 and 2, inclusive.
5. Similarly, we continue this pattern for each integer k until x = k. When k <= x < k + 1, g(x) = ⌊x⌊x⌋⌋ = ⌊x⋅k⌋ = ⌊x⌋ = k. Here, the floor function of x will always be an integer k.
6. Finally, when x = 10, g(x) = ⌊10⌊10⌋⌋ = ⌊10⋅10⌋ = ⌊100⌋ = 100.
By observing the above steps, we can see that the possible values of g(x) for 0 <= x <= 10 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 100.
Therefore, the number of possible values of g(x) for 0 <= x <= 10 is 12.
I hope this explanation helps you understand how to solve this problem. If you have any further questions, feel free to ask!