For homework we're supposed to give all the properties of different functions. I was able to do all of it except for this one greatest integer function -
f(x) = -1/2 [x-1]
These functions always mess me up!
Could someone tell me the following properties of this function.
Increasing Intervals
Decreasing Intervals
1-1?
Continuity
End-Behavior
Boundedness
Asymptotes
x-intercept (s)
y-intercept
Thanks I appreciate it a ton!!!!
The greatest integer or step function does require careful handling. You should graph the function f(x)=[x] and make sure you understand what the function looks like.
[x] on [-2,-1)=-2
[x] on [-1,0)= -1
[x] on [0,1)=0
[x] on [1,2)=1
Graph this for a few more values to understand the basic step function.
Now look at
f(x)=-1/2 [x-1]
f(x) on [-2,-1)=3/2
f(x) on [-1,0)= 1
f(x) on [0,1)=1/2
f(x) on [1,2)=0
f(x) on [2,3)=-1/2
f(x) on [3,4)=-1
Increasing intervals? Do you see any?
Decreasing Intervals Kind of tricky, but check that f(x) is monotonic decreasing on it's domain. I hope you're familiar with the terms.
1-1? If you can find two x values with the same f(x) value on an interval it's not 1-1.
Continuity If you restrict yourself to the interval [z,z+1) where z is any integer then it's continuous.
End-Behavior Is it continuous at the endpoints?
Boundedness Consider any interval I, it can be open, closed or mixed. Does f(x) have a min and max on I?
Asymptotes The answer should be self-evident if you graph a few intervals. Are there any vertical, horizontal or oblique asymptotes anywhere?
x-intercept Check where f(x)=0.
y-intercept Check the interval that contains x=0.
Be sure you know the basic step function. This function is widely used in economics and areas of applied math where we want only integer solutions.
Now check my calulations and explanations.
Ok I'm not sure, but then could you check my answers for these properties for the same function?
Increasing and decreasing intervals - None?
1-1 - No?
Continuity - No?
End Behavior - ?
Boundness - No?
Asymptotes - No?
x-intercept - (0,1/2)
y-intercept - (1.9,0)
Increasing and decreasing intervals - None? Correct about increasing, but technically this is a monotonic decreasing function. It's not strictly monotonic decreasing, so I'm not sure how your text is using the term decreasing.
1-1 - No? Correct
Continuity - No? Not continuous for the integers. Continuous for all other reals.
End Behavior - ? Discontinuous for the integers, because for an integer n from the right it is (-1/2)(n-1) and from the left it's (-1/2)(n-2).
Boundness - No? f(x) is bounded for every interval except any that has +/-infinity describing it. I see no domain was specified, so we should suppose the largest possible domain is meant and that would be (-infintity, infinity) and thus f(x) is not bounded on R. the reals.
Asymptotes - No? Correct
x-intercept - (0,1/2) No, Check the table I gave. What interval has f(x)=0?
y-intercept - (1.9,0) No " " " " Find the interval that has x=0 in it and check it's f(x) value.
Let's go through the properties again:
Increasing and decreasing intervals - The function is not strictly increasing or strictly decreasing, but it is monotonic decreasing on its domain.
1-1 - The function is not one-to-one since for any interval, say [1, 2), both 1 and 1.5 map to 0. Therefore, there are two different x values that map to the same y value.
Continuity - The function is discontinuous at integers but continuous for all other real numbers.
End Behavior - From the right, as x approaches an integer n, the function approaches (-1/2)(n-1). From the left, as x approaches n, the function approaches (-1/2)(n-2). These values give us the end behavior of the function.
Boundedness - The function is bounded within any interval, except for intervals that have +/- infinity as their boundaries. Since no specific domain was given, we assume the function is defined on the set of all real numbers (-infinity, infinity). Therefore, the function is not bounded.
Asymptotes - The function does not have any vertical, horizontal, or oblique asymptotes.
x-intercept - To find x-intercepts, we need to find where the function equals zero. From the table you provided, we can see that the function equals zero in the interval [0, 1). Therefore, the x-intercept(s) is/are in that interval.
y-intercept - To find the y-intercept, we need to evaluate the function at x=0. From the table, we can see that f(0) = 1/2. Therefore, the y-intercept is (0, 1/2).
I hope this clarifies the properties of the greatest integer function for you. Remember to always refer to the graph or table of the function to verify your answers.