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7r + 13 (1= Zc}(3) Find the domain of the variable (in Ghe interval form) in the expreeeion...

Question

7r + 13 (1= Zc}(3) Find the domain of the variable (in Ghe interval form) in the expreeeion

7r + 13 (1= Zc} (3) Find the domain of the variable (in Ghe interval form) in the expreeeion



Answers

Find the domain of the given function. Express the domain in interval notation. $$R(x)=\frac{x+1}{\sqrt[4]{3-2 x}}$$

We're being asked to find the domain of the given function. Well, I see we have a fraction here, which means our denominator can never be equal to zero. So we have to figure out Are there any values of X that will make it zero? So we'll three x ever be equal to zero. Well, this will happen if x zero, because three times zero is zero. So we know X cannot be equal to zero in our domain. However, X can be any other value, and we will have a defined value for our function. So what we will say for our domain is that we will have all really numbers. Except, like we said, X cannot be equal to zero. So our domain is all real numbers except

Considering the expression 1/3 minus X. Since we're divided, you can't divide by zero, and if X is three, Um then we're dividing by zero. So real numbers, except X cannot be three. An interval notation, if you're learning this would be from negative infinity to three. Not included, Union three not included to infinity. That's all real numbers except for three.

We are given that the function J f x is equal to Natural log of X -3 bikes. And we're supposed to find the domain of this particular function. So what you mean by domain domain is a set of values that can be plugged into a particular function into a particular function so that it gives us a result. Or it is a set of values worthy function. This particular function is defined. So it is a set of values. Now, for example, that comes to the function of log X. It just means since that text has to be greater than zero, that means log endemic functions are defined for any number that is greater than zero. There is any real number that is positive. When X is equal to zero or negative. It is not defined. That means X can be zero or negative when it comes to longer than the function. Now, in order to function, sorry, find the uh, domain of this particular function. What we need to do is that we just need to apply this. But this particular condition here itself That is X -3. My ex is great. Run in the room since now we'll just split the fractions. That means X by X by minus three by X. Which means one minus three by X. Rate of that little as you use us one greater than three by X. What's news? X has to be great three. So this simply means that X is greater than when X is greater than three. The function is valid for example, then looks sick when uh Access greater than three. So that means if it is accessible to four We'll get 4 -3. 4 -3 divided by For this means one x 4. This is a positive number for which log functions defined. But one another thing that you need to consider is that here this is a fraction right? X -3 divided by X is a fraction. The only condition is that this political faction has to be greater than zero. So let me just take an example. We give the value for X to be minus two. When X is minus two, this becomes a distraction, becomes minus two minus three, divide and light minus two -2 -3 -5, divided by -2. Just five x 2. Right. So this is also posting number that is greater than the role. Hence logjam is function is defined for this particular value also. So this doesn't Agree with the condition that we got here. This value -2 is actually less than three. Right? So this means that any value that is less than three can also be. Does this mean that any? Well that that is less than three can also be included in the domain of this function. No, it doesn't. It just means that See obviously uh three can be given for this value. Right? When you take three for example, uh So let's consider the number line here. When you you helped three here, you have zero here and you help Uh negative for example, negative three negative numbers here. So here what what according to this condition, what we have got is that we can take any number that is greater than three. Now, let's split this into the section two to a different sections. That is oneness. This section any number? Great country. The address any any number that comes between zero and 3. Okay then any number that is less than is it all? That is any number that is negative. Okay, that -2 comes in that. And so we checked with the number that is greater than three. That in this particular area we can take it right Because you can buy the domain exists. Are the numbers in this section exists in the domain? As we checked with four. Any number that is greater than 43 Now, coming to this particular radio when you take three, it is obvious that three minus three is zero with log log of 00 is not defined. Hence we can't take three. So let's say uh we take one accessible to one. What will be The number? That is 1 -3 divided by one with views minus to do what we want, which is -2. Which is not acceptable for it's like having function right? Hence we can't take this one other thing. Hence we are sure that we can take this particular interest. And then what about the other thing that you need to consider is that you need to make sure that our the whatever the values at the bodies that is 03 for this brilliant. Because zero can be if zero can zero is also included. Can be also including the in developed is left towards the room And three can megalodon was right towards the three. So when you can sit three we found that we can't take three because when we take three the filter and what we grow. So what about zero? When we take X equals zero. This fraction becomes zero minus three developed by zero. This means that we have a division developed by zero which is not defined which is also not possible. Hence here we can take any value that is included. Him. This political interval and this political interval that is any number that is less than zero and any number that is greater than three. And this particular function is valid for all that values. Hence the domain of this world really. Function is minus infinity to minus three. Sorry minus three. Yeah, yeah zero. But we can't take obviously we can't take minus 20 and also we can't take zero because when you take zero the division is not defined. Hence you have to give an opening to be here nineties into that. You also have any number that is greater than three that is three. Opening Trail 3 to Infinity. So I'll just go through steps once again. So you just have to do a bit of thinking here. I mean to get the answer. So the thing is you know what is domain and we applied the domain condition for longer than the function to find the domain here and we got that X is greater than three. But when you have a fraction you just need to think about a little bit. That means There can be something to stay here. Okay. So the first thing obviously it's obviously it's clear that X can be X has to be greater than three But then X shouldn't be less than three. Is what this particular thing says this condition says. But when we just checked it with the number that is less than three, that is -2. We found that you're getting a positive value for the whole fraction, which means that this is not the only interval which is and which can be controlled by domain. So then we just went on to check again with the number line will be divided into three sections. Well, with the role three, That is an interval numbers are less than zero. What are interval of negative numbers then? Or an interval of numbers are greater than 033 and then interval where it is in between the drawing three. Hence we just found out that the domain extends from minus infinity to open in 20 and then Open and 3 to Infinite. I hope the concept is clear for everyone. That's it.

All right. Welcome back. So for this problem where he had the function years, you equals Z divided by Z minus three. All right, so we can see that this function will be volatile, would rate when Z is equal to three. Because when Z is equal to three the denominator eight year, this stuff will be equal to zero. That is bad when we're dividing by zero. So we don't want that to happen. So we can say that just not acquitted. Three. Where we can also write this as this set from negative infinity to three. You're munis it from three to your infinity. Not that we're using parentheses right here instead of braces or brackets, because must so that we don't include the value three.


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