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The Floor and $C E I L I N G$ functions are defined as follows:$$\begin{array}{l}y=f \ln (x)=\left\{\begin{array}{c}x, \text { if } x \text { is an integer } \\\text { integer to } x \text { 's left, otherwise }\end{array}\right. \\y=\operatorname{ceil}(x)=\left\{\begin{array}{c}x \text { if } x, \text { is an integer } \\\text { integer to } x \text { 's right otherwise }\end{array}\right.\end{array}$$ Use these functions to solve.Sketch the graph of $y=\operatorname{ceil}(x)$ for $-3 \leq x \leq 3$.

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Missouri State University

Campbell University

McMaster University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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02:05

The Floor and $C E I L I N…

03:40

03:32

02:03

The floor function, or gre…

02:08

Ceiling function The ceili…

02:17

The ceiling function, or s…

03:02

Graphing Functions Sketch …

01:49

? Graphing Functions Sketc…

02:38

The greatest integer funct…

for this problem, we're going to be graphing the ceiling function the ceiling of X from X equaling negative three through positive three. Now, before we try to graph it, let's review what the ceiling function is. We've got two different cases we look at. We write. This is a piece wise function. First. If X is an integer, then the ceiling function just returns. X. The ceiling of two is to the ceiling of 10 is 10. The ceiling of negative eight is negative. Eight. If it's not an integer, then the ceiling function returns. The integer two X is right, so we're always moving to the right on these. So the ceiling. If I pick a number between one and two, let's say I pick 1.4. Well, that's gonna move to the next one up, so that's gonna move up to two. Okay, so let's try to graft this now, on this grid here just to make it easier to see every two spaces is gonna be a unit. So I've put out a couple of the numbers here. Just help us see the scale first. I know that if I have an ex as an integer. The ceiling function returns the same integer so I could mark all of the integers on this graph. 123 on. I could do the negatives. Negative one negative to negative three. Remember, again, every two blocks is one unit doing it. One and one makes it just scrunches our graph down. It makes it hard to see. Okay, now what happens between here? Let's go between zero and one. Well, you're always moving to the right. So anything between zero and one? Yep. Anything between zero and one is gonna have a value of one. So I haven't put an open circle there at the 0.1 because at when x zero, the value is actually zero. That's where it's dropped down. But everything else between X equals zero and exactly one has a value of one between one and two. Everything has a value of two remembers moving up to the next number between two and three. It's up a three. And that pattern continues down with my negatives, right at the integer it is. It's value. As soon as I get just a little bit to the right. It jumps up to that next value. So it's a stair step. Look is how what the function looks like. Both ceiling and floor functions have that one step, one step, one step. Look to it. It's just a matter of where the steps are in relation to exit changes for ceiling to floor. But this is the graph of the ceiling function from negative three to positive three.

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