if-fxxsqrt2-x-and-guusqrt2-u-is-it-true-that-fg-6

Name: if fxxsqrt2 x and guusqrt2 u is it true that fg 6
Uploaded: 2022-05-17
Channel: Numerade
Description: If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Question

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Logan Henry · Accepted Answer

Alright. So we&#x27;re given two functions F of X and G. Of you and ask, are these two equal to each other? So how do we determine that two functions at their basic most basic definition are equal if both of their remains are equal. And then if for those, that domain, if both functions F of X and G of you are equivalent have equivalent outputs for every single input. And so I&#x27;ll show you what that means in just a sec. But let&#x27;s check the first part of that. Roll out. Are there domains equal? And so let&#x27;s take a closer look at the Fedex. Specifically we&#x27;re told F of X is equal to X plus the square root of two minus X. So looking at the equation, what is our domain? Well, we&#x27;ve got a radical here and so we know we&#x27;ve got to take a closer look at that. What goes under the radical needs to be greater than zero because we can&#x27;t have any imaginary numbers in our function. And so adding that X over to the other side of the inequality, We can determine that X needs to be less than two or that our domain is going to be all real numbers less than two. Cool. So let&#x27;s do the same thing for our other function G. Of you. We&#x27;re told that G. Of U. Is equal to U plus the square root of two minus you. So, again, we&#x27;ve got another radical. And so we know that the expression under the radical to mine issue needs to be greater than zero and still doing the same thing we did with ffx. We determined that too needs to be less than you. And so our domain is going to be again all real numbers less than two, awesome. So we determined that both of our domains are equal right? Because they&#x27;re both equal to this are all real numbers less than two. So that&#x27;s half the battle right there. But next we need to determine for that domain for all real numbers, less than two do F of X and G of you have equivalent outputs for every single one of those points. So first let&#x27;s just go ahead and plug in a random number that fits here. And so I&#x27;m going to try um X and U equals zero. So let&#x27;s plug that into our equations. We have F of X is equal to X plus the square root of two minus x. And so if we plug zero in there we get zero plus the square root of two minus zero, Which is just equal to the square root of two, awesome. So let&#x27;s do the same thing but with G. Of you. And so again, I&#x27;m going to set that argument equal to zero. And so we nog of U is equal to U plus the square root of two minus you. And then I&#x27;m going to plug into zero wherever I see a you and what do you know? We end up with Route two for both problems but I know what you&#x27;re saying. Okay so they F. Of X and G. Of you are equal for one value of the domain when X and you both equal zero. But how do we check it for every single value? Because that&#x27;s how what we really need to be doing here. Right? And so I want to show you what happens when we just look at the functions with um function variables themselves. So rewriting F of X. You know we&#x27;re told if index is equal to X plus The square root of two minus X. and so now just like we did with the with the value zero instead of Reprint X0, I&#x27;m going to input you as my argument. And so everywhere I see an X I&#x27;m going to plug into you instead and you can see when I do that, I get an identical expression as to the one that were given for G. Of you in the beginning of the problem that you plus two root minus two minus you. And so doing the reverse of that, I&#x27;m gonna plug X into G of you. And what do you know? We get the same exact expression that we were given for F. Of X in the beginning because G of you as you remember, is equal to U plus the square root of two minus you. And when we plug an accent throughout the, you know that that equation, wherever that argument pops up, we end up getting the exact same equation. And so knowing that you and X both have equivalent domains, that means that whether or not it&#x27;s a U. And X. Whatever variable you plug in both equations F of X and G of you are going to be identical. And so the answer to our problem is yes F of X is equal to G. Of you, mm hmm.

Heather Zimmers · Answer

here we have f of X, and we have G of X, and we want to know if they&#x27;re equal. So chances are what you&#x27;re going to do is look at f of X and realize that you can factor the numerator. You can factor in X out of both terms and you have X Times X minus one over X minus one and then you&#x27;re likely to think, Well, why don&#x27;t I just cancel the X minus ones? And then I&#x27;ll have left his ex and it&#x27;s just f of X equals X? Well, that&#x27;s only partially true. Go back to the original function and think about the domain. Let&#x27;s think about the domain of F of X. You can&#x27;t have zero for your denominator, so that means that X cannot be won. Let&#x27;s think about the domain of G of X G of X is just a polynomial function, and all real numbers can be and put it into their, and an output of real number will result. So the domain of G is all real numbers. So from that piece of information alone, we know that they&#x27;re not the same function. And if you want to consider the graph, consider the graph of G. It&#x27;s just going to be the line with a slope of one and a Y intercept of zero. So then how would the graph of F B any different? Well, since you can&#x27;t have X equals one, you can&#x27;t have the 10.11 that would be on G. And so you would have a whole at 11 And other than that, you would have something that looks like G. So these are not the same.

Nandini Singh · Answer

number two says that if F of axe equals X squared minus acts over X minus one and g about people&#x27;s acts, is it true that pickle juice the first thing you do is simple by half a box? How you can do that is on the topic in fact row. Yeah, X and then we get X minus one. Excellent. It&#x27;s one, and you can cancel. Excellent is on the top of the bottom. But whenever you do that you need displaced by the ex cannot equal one and then you&#x27;re left with just fax. So you can come to a conclusion that after sex equals g a box on the interval, I give her affinity told one non inclusive or one part of divinity. Reason being is GI effects exists on the domain of all real numbers. But effort backs exists on the domain of all real numbers except when X equals one. So you need to specify that because, uh, X equals one F of X will be undefined. Reggie of X will have a point. So at that point they will not be equal to each other, but all other points on the curve they will be equal to each other

Renée Przybyl · Answer

and this problem, we need to decide whether or not F of X is equal to G of X. We can do this by simplifying Methanex. You see there&#x27;s a common factor. X in the numerator, said X Times X minus one over X minus one. We could simplify that f of X is equal to X now this makes it seem that f of X is equal to G of X. But that is not the case because we have to go back to F of X and notice that the domain of F of X is going to be different because of the denominator. So X minus one cannot equal zero because it will make ffx under find. So that means X cannot equal one. Now the graph of F of X is going to look like this. And since X cannot equal one, there is going to be a hole in the line. Well, the domain of G of X is X can be anything from night of infinity to infinity and its graph is going to look like this. Now we see that these two graphs are different because there is no hole in G of X Well, there is a hole in f of X due to having the possibility of being undefined. So this means that G of X and ethics are different, so they&#x27;re not equal.

Subhadeepta Sahoo · Answer

you discourse. Senator Command that f off accepted Contador&#x27;s attempts on G o X X squared plus one that makes bloody off, actually portal for thanks, the TF X. It affects less square last one. Thanks very well. That&#x27;s what that&#x27;s picks unless one no explain next year. Next to sex. My fix. Place minus X squared. My next one minus picks My six man. It&#x27;s in June, minus F off exit people. Put your fix my aftereffects next. That that&#x27;s what. No thanks, my missus Ethics.

Amy Jiang · Answer

Okay, so we&#x27;re given back and f capos g of that&#x27;s that&#x27;s equal to two acts. So that&#x27;s it seems like physical to this term here. So what&#x27;s Dio? So we basically need to counter out this term here. That means gov is equal to extreme power of three. So if we have f composed of G back, you know that you&#x27;re back. That&#x27;s equal to execute in an F supplied ads X. Actually, we should have a two here. Does he still want that, too? Um, who gets the Cube roots of two extra factory and that gives us two X, which is what we want here.

Logan Henry · Answer

all right in this question were given to functions at the vex and also G of X. In this first function of X is equal to X plus root of two minus X and homes are in here. Mistake. It&#x27;s not she of experts, actually she of you in this function, she of you is equal to you, plus route to minus you. And the question that were asked is if f is equal to G and when we look at both the functions f N g, we noticed that they use X and you as a variable names. But besides the variable days being different X and you, everything else about the function is exactly the same. It&#x27;s just a variable added to the square root of two minus that variable. And for all intensive purposes, these two functions, then, are exactly the same because of the fact that these variable names can really be anything. We could call this you. We could call this X, and there would be exactly equal to one another, so f equals she in this case

Logan Henry · Answer

Okay, So we&#x27;re asked if f of X is equal to X plus square root of two minus x A and G of you. Is he put to you that skirt boots of you? My nephew? Is it true that definitely quitting, G. What is true? Because all we need to do for FX. Let&#x27;s say we start with that vex and we change our f too g and then our exes to you. And they&#x27;re basically the same value. So, yes, this is true.

Logan Henry · Answer

when you have functions like F of X and G of axe and H of X, and so on. The letter you choose to use for the input is really arbitrary. There&#x27;s nothing sacred about using X. There&#x27;s nothing sacred about using you. So this function here, which is currently called F of X. It&#x27;s the same function if I call it f of you. And if I replace all the exes with use, it means the exact same thing. And this function g over here, it&#x27;s the exact same function. If I replace all the use with exes, so are these the same function G of X equals X plus two plus the square to minus U f of X equals X plus square to minus you. Yes, those are identical.

Logan Henry · Answer

Therefore X equals two X plus square root of two minus x. This function is defined only when 2 -1 is greater than or equal to zero, which gives us X less than or equal to. So the range of F function F is minus infinity to the second function G of U equals two U plus square root of two minus you baird two minus. You should be greater than or equal to zero. For G to be defined, which gives us you should be less than or equal to two. two functions FNG are said to be equal only when the domain and co domain are seems so F is equal to G only when X comma, you belongs to minus infinity to two. So for all values of X comma you in the range of minus infinity to two, F is equal to G.

Logan Henry · Answer

In this question, we have two functions F&G but be careful in the function f. We use the variable acts you see X here. And in the function G we use the variable. You are Right, Okay. So uh variable is not really important for the functions, you can change the variable, You can use any letter you want for any function. So for the function F of X which is given in the question as um X plus a squared of two minus x. Uh in the in the question also gov function function G is given in terms of the variable U and U plus a squared of two minus you. So I can change this letter this variable by another letter. So I can rewrite G o of you as U of X, which is just substitute you with X. So it can be written as X plus a squared of two minus x. So if the domain is the same for two functions, so we can get fo fax is equal to geo fax. Which both functions can be written in terms of X as X plus squared off two minus X. So we can easily say that F and G functions are equal to each other, so F is equal to GG

Problem

If $ f(x) = \frac{x^2-x}{x-1} $ and $ g(x) = x $ …

Problem 1 Easy Difficulty

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Answer

True.

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