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F(t) =2+2+7 tan-!z has exactly one Froblet Prove tbat the function...

Question

F(t) =2+2+7 tan-!z has exactly one Froblet Prove tbat the function

f(t) =2+2+7 tan-!z has exactly one Froblet Prove tbat the function



Answers

Proof Prove that a function has an inverse function if and only if it is one-to-one.

Still, we're given the function, and we're actually show that it's even so First, we look at a definition off uneven function, which is a function that satisfies up of negative X is equal to every day. So what we're gonna do is to check it. Our function satisfies dis equation. So we go ahead and put an effort negativity into the equation. So where we see on attacks were going to put a negative x Winstead. So we're good negative x into the equation wherever we see X, and then we separate the power because I'm doing this will get closer to making it look like original function. So I'm going to write it as negative x squared with apparent Aziz to the power and because two times and is going to give you to it. So you split it up into two factors which will not apply to give you the original. With a 2nd 1 you pull out, too, and what's left of worries and minus one. No, class that and this one stays the same, says there's nothing to factor in the power. So now we see that we can square the negative X to get on eggs because any number squared is going to give you a positive number so we can get rid off the parentheses and haven't x squared to the end That a to n ah ah, The second term is supposed to be a to the two and minus two. So two and minus two. And that should be good. And with this one, have Julianne ex left over last night. So now I'm going to simplify the power of the second term because two times and his two and in two times one is two. So we simplify and looking at it, we see this is the same and fo fate, so therefore it's even.

To be 11 f off. If if f or physical therefore be, then it would be also equal. Toby No other function. F X is Cuba. It affects. So for this if f off a So if a four phase it will do a for B. We can say that Que brute off phase you could cube root off beat after cubing on both sides will get a sickle, Toby only so we can say that if ever, for a single drop for his also equal Toby so f or faxes 11

Okay, so we have a proof over here we're going to touch. It proved that if a function has an inverse than that inverse is unique. So we're gonna do is something called approved by any version. So which is basically saying that we're first going to assume that the universe function of F is not really so. We're going to define to functions, let's say F one and f too, that are both in verses off so that no in verse is equal to one. And as Inler's is also equal to ft. Next we're gonna let a equal the range of us. Since that one off eight is equal to inverse of A is equal to be one which implies that half of the one is equal to a and similarly since there also exists an F two which follows the exactly in principles we have afternoon, eh is equal to ask inverse of a which is equal to somebody too such that F B two is also equal. Ta. Now this right here is our contradiction statement. So this means that is not a wonder one function since it says that F B one is equal toe F b, too. Where be one does not equal be too. Thus does not have an inverse function. So therefore, we have proved by contradiction that the inverse function has to be unique. So this proves that the inverse of F is unique because it is a wonder one function. And this means that every element in the domain of F has only one in the UAE. All right, that's all.

If F is a 11 function, we want to show that G of X equals negative fx is also 1 to 1. So remember that if f is 1-1, then that means that F of X one Equalling Half of x two implies the X one equals X two. So, um now we want to consider G fx, which is a negative fx. So we're going to assume That um negative f of X one equals a negative affects too. And we want to show that X1 still equals X two. So that's the method that we take. We know that this is true because and we could divide the negative one and get back to this original form. So this holds is a true statement. Mm


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