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(b) (4 PTS) bijective. Give a domain and codomain for which 34 function f:Af-1({2}) The f-'([-1,0]) f ([o,;]) H (a) (6 PTS) Find the following cosx is function...

Question

(b) (4 PTS) bijective. Give a domain and codomain for which 34 function f:Af-1({2}) The f-'([-1,0]) f ([o,;]) H (a) (6 PTS) Find the following cosx is function f: [ given at the right: [uz 21t ,defined by f(x) cos*is

(b) (4 PTS) bijective. Give a domain and codomain for which 34 function f:A f-1({2}) The f-'([-1,0]) f ([o,;]) H (a) (6 PTS) Find the following cosx is function f: [ given at the right: [uz 21t , defined by f(x) cos*is



Answers

(a).Sketch a eraph of the function $f,$ (b). Determine an interval on which $f$ is one-to-one, (c).Find the inverse function of $f$ on the interval found in part (b), and (d) give the domain of the inverse function. (Note: There is more than one correct answer.) $$f(x)=3 \cos x$$

Okay, so he are given function is cause function, and we need to used the derivative test toe, determine that the function is strictly monitor funny or not. So if it's derivative with either less than zero or greater than zero, then we can say that this function is strictly monitor Nick on its own domain. Right. So here, first of all, we need to take very with you now derivative off course function that is equal to negative sine function. And here, using the general, we need to take very witty off its angle. And here angle is three excavated by two. Now they re video off three x by two. That is a call to three by two. So you're in financial. Fax is a call to negative three by two. Sign off three. Explain to right. So here, if this affects can be less than zero on, it can be better than zero night because if we draw a unit circle like this and he are vertical axis that represent the sine function, right, so he are sine function is positive on its 1st and 2nd quadrant, and here sine function is negative on 3rd and 4th convent. Right? So you're after this off axis last 10 0, or it can be better than zero. Right? So here function is not monotone IQ. So he are. That is not Moloch tonic. Therefore, the function F or fax has no Inglis. Right? Thank you.

This problem gives you F of X is equal to the square root of X plus four. So the square root of X plus four. It wants us to evaluate for F zero F. Of negative three, negative four. Negative five. So let's plug those in and see what we get. So the square root of zero plus four just equals the square to four which equals two. The square root of negative three plus four is equal to the square root of one which is just equal to one. The square root of negative four plus four is equal to the square of zero which equals zero. And the square root of negative five plus four equals to the square Square root of -1. Which is not possible, does not exist as a real number. So we know that there's gonna be some domain restrictions there and that's what part B is asking. So remember the square root, we must be taking a number that is zero or greater. Right? So This X-plus four um Must be greater than or equal to zero. You can take the square root of zero. So solve for X by subtracting four from both sides And you get that X must be greater than or equal to negative four. So meaning the domain for X, then it can be equal to negative four. Um And it's got to be greater than that. So that'll be the minimum is negative for and I can go all the way up to positive infinity.

I recommend domain of this. We have been of course in the world and then of course and what we have that's causing more text we have domain is given as a negative one has an equal to X. Is equal to the domain of ocean vortex. You ready for this? We get here. He's coming out of a negative one less than Ippolito. Logo X over paul. The best three has an equal to one. Now if we just sold this here. So where does negative one less than what we have the logo X over port or we have Lot of 3: one. So we get here negative a lot three less than equal to logo acts or fall less than equal to Vienna logo 30 board here. Now all in laws when you get here, bodies look pretty about negative one lesson in court of law and our poll, I wasn't able to we have logo tweets, we get rid of a negative one. Less than equal rights over four. Less than equal to scream. We get from here that's coming up with 4/3. There's an equal to act less than equal to. Well, the domain is coming out to be For over 2012 and close intervals there are three options. Deep look Andy thank you

For the given problem. We want to sketch a graph of the function. So this is gonna look like f of X equals the square root of x squared minus four X. And we see we just want to restrict the domain in such a way that we can find the inverse. So we're going to restrict this. Um such that X is greater than or equal to four. So just this portion of the graph. Now we can rewrite this in such a way, why is equal to it? Um So when we square both sides and then we uh we score both sides and we want to solve for X. So we end up getting um this right here when we end up solving, we'll have minus four, will take out the X. So says and four. And then we will we're gonna want to use completing the square. So we're gonna end up with the square root Okay. Of wide squared X squared quest for secure. And that will end up being again restricted but no longer from X. Greater than or equal to negative for but why? Greater than equal to four. So this is what we end up getting. Because why is equal to this? And this is going to be zero or X greater than or equal to? Yeah, that's the final result.


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