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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.Determine a relationship between $f r(x)$ and $\operatorname{ceil}(x)$.

For non integer values, $\operatorname{Ceil}(x)=\operatorname{flr}(x)+1$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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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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The Floor and $C E I L I N…

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match each function with i…

for this problem, We're going to just look at these two functions the floor function and the ceiling function. How are they similar and how are they different? Well, you can see the definitions of the Florence ceiling functions right above exercise number 76 and you'll see right away that they do have something in common. If X is an integer, both the floor function and the ceiling function return X. So if X is an integer, then these two functions are equal to each other. Floor of X is gonna equal the ceiling of X. Sorry, that's rather hard to read. Let's try that again. Ceiling of X. So, in other words, if I put in five while the floor of five is five, the ceiling of five is five. For an integer. They are identical. But what if they are? What if it's not an imager? That's where they differ. If it's not in a major, let's say I take a number like 3.1. Well, the floor function to the floor function here in the ceiling function here, the floor function moves it down to three. The ceiling function moves it up to four. How about 6.8 floor function says you drop down to the left to the left side of that number. Sailing functions as you move up to the right. How about a negative number? Let's say I have negative 2.6. Uh, the floor function says. I moved down to the next number on the left, which is negative. Three Sailing function says I moved up to the right, which is negative. Two. And if you look at these numbers here and compare them, you'll see that if X is not an imager, the floor function is always one less than the ceiling function. So we can say that if X is not an integer than the floor function, is the ceiling function minus one?

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