(1 point) Let f(x) = lfloor x/2 floor. We learned that the...

(1 point) Let $f(x) = lfloor x/2 floor$. We learned that the floor and the ceiling functions are NOT invertible, but we also learned about the set of preimages of any value in the Range, the set of images. Keeping that in mind, give your answer in interval notation if necessary. (a) Find $f^{-1}({3})$. Your answer is (b) Find $f^{-1}({-5})$. Your answer is (c) Find $f^{-1}({x mid 3 le x le 6})$. Your answer is (d) Find $f^{-1}({x mid -6 le x le -5})$. Your answer is

Transcript

00:01 All right, so in this question, we're looking at f of x equals the floor of x over 2.

00:06 And so remember what that means is you're taking the sort of least positive, or the smallest integer less than x over 2.

00:14 And so if you plug in 0, 0 over 2 is 0, and anything all the way up until, right, if you plug in 1, you get a half.

00:25 So the first integer less than a half is still 0.

00:28 So you're still 0 there.

00:32 But as soon as you hit positive 2, then you get 1.

00:35 So at positive 2, you're up here at 1.

00:40 So what we have is this type of behavior.

00:46 So then, again, when you go to 3, 3 halves, the floor of 3 halves is still positive 1, but then as soon as you hit 4, 4 over 2 is 2, so you jump up here to 2.

00:58 So you get this sort of stair step -style function, where each blue bar is width two, then each vertical jump is only worth one, height one.

01:16 And this function doesn't have an inverse, but you can take pre -images.

01:22 And so we're asked a few different things.

01:24 We want to find the pre -image in part a of just the set three.

01:30 So what that means is we want to find three in the range.

01:38 Right, the y value 3, we know what are all of the input values to this function that could give us 3 as an output.

01:46 So let me just go ahead and draw that next little segment on.

01:53 What you see is sort of the first value they can give us a 3 is 6, right? because when i put 6 in, i get 6 over 2 is 3.

02:05 So i know that if i plug 6 in, so i'm going to use interval notation here.

02:11 All of answers are going to be in intervals.

02:13 So i use interval.

02:14 It's a bracket because i can include six, right? like if i plug six in, i definitely get three out.

02:20 You can go all the way up until you hit eight.

02:26 But we can't include that, right? because if i actually included eight, eight over two is four.

02:31 So then you're jumping up to your output values being four.

02:33 But any input values between six and eight, when you take the floor of those values divided by two, you'll get a three out.

02:41 So the pre -image of the same.

02:42 Single element 3 is a whole range of values, right? it's a whole intervals worth values.

02:50 So for part b, we want the pre -image of the set containing negative 5...

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