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

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

University of Michigan - Ann Arbor

Idaho State 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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03:40

The Floor and $C E I L I N…

02:39

02:03

The floor function, or gre…

03:32

02:08

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…

04:05

Graph each piecewise-defin…

for this problem we're going to graph Why equals the floor of X for X values between negative three and positive three. Inclusive. Let's take a step back and review the floor Function well if the floor function, we're breaking this up into two pieces. First, if X is an integer, then the floor function just returns X So the floor of three is three. The floor of negative five is negative. Five. Okay, so floor function oven imager is just that imager. But if it's not an integer, we're going to return the integer two x is left. Okay, so we're always going down to the next integer below the value that we've put in. So the floor of 3.1 shift down to three. So let's take a look at what that looks like when we graph it first. We know that the floor of an integer is the integer, so I can mark all of those points already on my graph. I'm just going from negative three to positive three. So that gives us those seven points. The floor of 001 is one too is to and so on. But what about the numbers in between. Well, we know that we're always going to the left. So let's take a look at a number. Any number between two and three. Any number between two and three is gonna go down to two. I'm gonna put a little open circle there, because at three, it shifts up to three. But everything between two and three has the value of two. Everything between one and two shifts toe one. So I've got an open circle, and then everything from 1 to 2, um, has a value of one, and you could see these. This floor function is a step step function all the way down. Okay. On the integer, it is the point, everything else and that next one block has a flat value, and then it shifts up to the next one. So this is the graph of the floor of X from negative three to positive three

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