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Problem 31 Hard Difficulty

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ \displaystyle M(x) = \sqrt{1 + \frac{1}{x}} $

Answer

so $M$ has domain $(-\infty,-1] \cup(0, \infty) . M$ is the composite of a root function and a rational function, so it is continuous at every number in its domain by Theorems 7 and 9

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Video Transcript

Okay first we want to find the domain of this function. So let me make sure you understand what domain is first. So let's talk about just the square root of X. So when you're finding the domain of a function, what you're trying to find is what kind of excess are okay to put into the function. What kind of excess will give Uh real numbers back for the answer. So you can put forward in there and you will get to you could put 400 in there, you can put 401 in there, you could put one half in there. Okay? Even though they don't come out to be whole numbers or even if they're not perfect squares, you can still find their square root. Your calculator can mm Okay, you can put zero in there because the squared of 00. Okay, but what about negative for can you put negative for in there? Well the Square 2, -4 is imaginary. It's too I so you can't put negative for in there because we only want real numbers to come out again. So the domain of this function Is all X greater than or equal to zero. And if you look at the graph of it goes to 001142. So you can see that when I drew it, I only used X. Is that were greater than or equal to zero? So that's what domain means. What can you put in there? So in our function we have the same rule to use we have to make sure that this stuff underneath the square root is positive or zero. Okay, so one plus one over X needs to be greater than or equal to zero. Okay, these are just remember kind of tricky to solve. So don't be moving stuff and cross multiplying and all that kind of stuff because of the inequality sign. What you need to do is get a common denominator speaking at them together. So multiplied by X. Here. So you get X plus one over X. We need that to be greater than or equal to zero. All right then you say what will make the bottom zero? That would be zero. And what would make the top zero? That would be -1. All right. So we've broken up the real numbers into three sets. The number is greater than zero. The numbers between zero and -1 and the numbers less than -1. So now we've got to figure out which of these Numbers make sense in this function. So pick a number bigger than zero. Like five and then plug it in and you get five plus 1/5 is greater than or equal to zero. Okay, that's true. So that means all of these numbers work. Okay, pick a number between zero and minus one. Like minus one half minus one half plus one over minus one half. That's 1/2 over -1 half. Greater than or equal to zero. That's false. That means these between 0 -1. Are not good. All right. Pick a number less than minus one like minus two so minus two plus one over minus two. As -1/-2. Greater than or equal to zero. That is true. So these will work also. Okay what about my minus one? But if you plug minus one in there you will get minus one Plus one over minus one. That's zero which is greater than or equal to zero. Okay so we can use -1. Okay, what about zero? We get zero plus 1/0. Oh that's bad. So we can't use zero so we'll have to put an open circle there. All right. Now we can just write the answer down. I don't know how you're supposed to write it. Will write it in inequality notation X is greater than, sorry, That's his last Enrico -1 because that's worth over here Or X is greater than zero. So that's the domain. The numbers that satisfy these conditions are the numbers that make sense in that function. But the numbers that don't satisfy that, the ones between -1 and zero, they cause something bad to happen. And so we're not able to take the square root. The bad thing that they caused happen they make the underneath negative and you can't take the square root of a negative number. Yeah. All right. The other thing was explain why this function is continuous everywhere in its domain. Okay well zero is not in the domain. So we know that division is continuous or that one of her exes continuous. We know that if we add one to it, it's still continuous. Here is the square root function. It's continuous. So this is continuous everywhere in this domain, because each piece is continuous and every operation uh huh keeps the continuity. Okay. So we're not doing anything weird that will put any holes in it or will cause it to break or anything. So since square root and one over eggs and adding are all continuous, this function is continuous everywhere in its domain. Okay.