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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 $y=f \operatorname{lr}(x)+x$ for $-3 \leq x \leq 3$.

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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).

03:18

02:05

The Floor and $C E I L I N…

02:39

02:08

03:31

02:03

The floor function, or gre…

03:32

Ceiling function The ceili…

02:17

The ceiling function, or s…

03:02

Graphing Functions Sketch …

01:49

? Graphing Functions Sketc…

01:13

Using a chart of values, g…

for this problem. We want to graph the equation. Y equals the floor of X plus X, and we're gonna be looking at X values between negative three and positive three. So let's take a moment and review the floor function. The floor function has two pieces to it. First, if X is an integer, then the floor function of X is just X floor. Function of two is to the floor function of tennis 10 and so on. It effects is not an integer. Then the floor function tells us that it is the integer Toe X's left. It's always going below. So the floor function of 2.8 goes down to to the floor function of, uh, 10.5 goes down to 10, right? So let's take a look at what we have. And I'm gonna start with the integers because I know exactly the value of X when, x when the value of the floor function of X when X is an integer. Okay, so if it's an integer, this just equals X. So why would be to X And to make this easier to see, I am going to make every two blocks B one unit to spread out our graph just a little bit. Makes it easier to read to negative three. Negative one. Negative to negative three. Okay, so let's put some dots on here. Zero zero, because back zero twice X is still zero effects is one. Twice. That is to if excess to twice that is four. And if x is three, that's gonna put it up there to six. Okay. So that those points and I could go down and do the same thing here. Negative one is negative. Two negative two is negative. Four and negative. Three negative. Six. I just realized I put the wrong dot up here. Um, race that one up there. I didn't go up high enough. Three should go up to six. Okay, so those are my dots when x is an integer. Now, what effects is not an integer? Well, let's look at the area between zero and one through zero and won the floor function of all of those numbers. Zero. So all I have is X. So that means I'm gonna be going just like this. But when I get toe 11 I have an open circle because That's where it jumps up to the 110.12 Now between one and two. Uh, the value. The floor function is one. And I have, um, ex. I'm just having X. I've got X Plus one that's going to give me if you think about why equals X Plus one that's still going to be a slope of one. So it's going to come up this way, and I have That is an open circle. I did not make it very clear. Let's make that a nice open circle, okay, between two and three I have that's gonna have a value of choose. This is going to be X plus two. But again, it's the same slope and you'll see that all the way around. For all of these, the slope is one. I'm just adding a different number to it, so I get there still kind of that step motion that we saw when we graft just a plain old floor function. But now, since I'm adding something to it, they're like slanted steps. Um, because I'm adding Justin X. Each one of those steps has a slope of one. So this is the graph of the function. Why equals the floor of X plus X

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