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

True.

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Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 1

Four Ways to Represent a Function

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Xingtao Y.

December 16, 2017

What if u is equal to one over x?

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June 29, 2020

All of these questions are answered for me... IS there anyway for me to answer myself?

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October 17, 2020

David Base G.

October 27, 2020

Finally, now I'm done with my homework

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This will help a lot with my midterm

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May 20, 2021

does it matter if you use different variables?

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July 23, 2021

At 0:37 did you mean 2-x instead of 2-u?

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If f(x)=x+

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Not always; it depends on their domains. Please check my explanation!

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Ok

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A multivariate function is…

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Alright. So we'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'll show you what that means in just a sec. But let's check the first part of that. Roll out. Are there domains equal? And so let's take a closer look at the Fedex. Specifically we'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've got a radical here and so we know we've got to take a closer look at that. What goes under the radical needs to be greater than zero because we can'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's do the same thing for our other function G. Of you. We're told that G. Of U. Is equal to U plus the square root of two minus you. So, again, we'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're both equal to this are all real numbers less than two. So that'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's just go ahead and plug in a random number that fits here. And so I'm going to try um X and U equals zero. So let'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's do the same thing but with G. Of you. And so again, I'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'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'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'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'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'm going to input you as my argument. And so everywhere I see an X I'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'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'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.

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