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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.(a) $f(5),$ (b) $f \operatorname{lr}(-5),$ (c) $f(5.1),$ (d) $f \operatorname{lr}(-5.1)$

(a) 5(b) -5(c) 5(d) -6

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

McMaster University

Harvey Mudd College

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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03:32

The Floor and $C E I L I N…

02:08

03:40

02:39

02:05

02:58

The functions are all one-…

02:46

A function $y=f(x)$ satisf…

02:15

A function $f$ is given by…

03:22

For the following exercise…

04:45

Determine whether the answ…

for this problem. We're examining to new functions, the floor function and the ceiling function. So let's take a moment and see how these air defined in both cases. If X is an integer, both of these functions will just return the same integer. So if X is an integer for the either the floor or ceiling function, we just return X. Nothing changes. What's different is if X is not an imager. The floor function says we're going to take the integer two x is left. If it's not an integer, if it's a ceiling function, we take the integer to the right. So okay, that looks like night. Let me let me right, right that So it actually says what I said. It says integer to the right. So let's take a number line. And if I put on, I'm just gonna put maybe three or four numbers on here. I've got 123 and four. Let's say I have pi 3.14 It's not an integer. It falls between three and four, so the floor function says I'm going to go to the next integer to the left so the floor function would return three ceiling function says we're going to go to the next imager to the right, so I'm going to go up to four. Ah, good way to visualize. This is to put it as if it really is a floor and a ceiling, and I'm going to stand the number line on its head is we're gonna pretend these air floors, maybe on a hotel, one first floor, second floor, third floor and so on. If I put pie here again at 3.14 the floor function says it's going to sink to the floors. Was going to go down. That puts it down to the three. That's the equivalent of going left on the horizontal number line. The ceiling function says, is going to float up to the ceiling. That's the equivalent of going right on the horizontal number line. So let's actually try to find some values here. Um, first, let's start with some integers floor function of five, right? Well, the floor function of five means I'm gonna sink, but I'm already an integer so it doesn't change. Four function of five is five. The floor function of negative five is negative. Five. That's also an integer integers don't change your already right there on the floor. You dont have to rise or fall. What changes, though, is if it's not an integer. So let's find the floor of 5.1. Well, 5.1 again, If I'm on this little imaginary hotel is right there above five, Floor says, I'm gonna sink to the floor below, which means that it's going to go down to five, which makes sense. We're just going down. We're kind of truncating that end. It's important, though, to look at what happens for negative numbers. So I have negative 5.1. It's very tempting to say, Oh, I'm going to truncate It goes to negative five, but it doesn't take a look at this and I'm gonna get I'm gonna do the number line this way. So let's say negative three negative four. They get smaller as they go down, and negative 5.1 is actually below five. So the floor function says it's going to sink to negative. Six is always to the left of the number that we're taking putting into our floor function. So here's the value for four different inputs into our floor function

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