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Sketch the graph of $y=\operatorname{ceil}(x)+x$ for $-3 \leq x \leq 3$.

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

McMaster University

Baylor 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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Sketch the graph of the in…

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for this problem, we're going to sketch the graph of the function. Why equals the ceiling of X plus X? We're gonna look at all X values from negative three to positive three. So let's take a moment and review the ceiling function. Remember, we split the ceiling function into two pieces. If X is an integer and the ceiling function just returns X, the ceiling of 12 is 12. The ceiling of negative 18 is negative, 18 and so on. If X is not an integer, then the ceiling function returns. The integer two X is right. We're going up to the next integer, so the ceiling of 1.5 moves up to to the ceiling of 12.9, moves up to 13 and so on. So when we go to graph this and you'll notice on my graph, just to make it easier to see every two blocks is one unit. So I've just stretched our graph a little bit here. Let's start with X being an integer because if exit integer in the ceiling of X is just X, which means that why would equal to X in those cases? So that means I'll have the point. 00 12 24 36 I'm dis doubling exes every time and three six. Okay, so those the points when x is an integer Now let's take a look When x is not an integer let's just do our first interval from zero toe one. So if X is somewhere between zero and one thin, the value of the ceiling function moves up toe one. So what I really have here is the ceiling function is going to be one. So I get X plus one. So that is a line slope of one and the y intercept of one. Now it gets a circle at that y intercept because the value and X zero is actually zero. That's where the jump is. Then it goes all the way up to the 00.12 Now what about between one and two? Well, then the ceiling, uh, function returns to it goes up to that next higher integer. So my function is really X plus two. Now it's the same. Why is the same slope? Just if you imagine the Y value of the Y intercept starting a two, that means it's going to kind of you can think about it coming up that way whenever you remove those, because they're not really part of our graph. We start here and we go up, okay, Slope of one. So it's got that same slope to it. It just had a different starting point. How bout between two and three? Well, now the ceiling function returns three hoops. Those exposed to now I get why equals X plus three. So again, same slope. I just have a new starting point, so you can see if I keep going with this concept here, Um, it keeps looking the same way. It just these lines go on a slant. They're slanted instead of the straight lines that we saw when we just did the ceiling function. Because we're adding the X to it, it puts them on a slant as a slope of why equals one all the way through. So this is the graph of our ceiling function, plus X

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