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Consider the mapplng f : R + (Rt 5 U {0}), where f (x) = x?, (2.5 points cach) Cetermlne If f Is one-to-one_ Justify your answer.Determlne IF f Is onto. Justlly vou...

Question

Consider the mapplng f : R + (Rt 5 U {0}), where f (x) = x?, (2.5 points cach) Cetermlne If f Is one-to-one_ Justify your answer.Determlne IF f Is onto. Justlly vour answier

Consider the mapplng f : R + (Rt 5 U {0}), where f (x) = x?, (2.5 points cach) Cetermlne If f Is one-to-one_ Justify your answer. Determlne IF f Is onto. Justlly vour answier



Answers

Suppose that $f(0)=5$ and that $f^{\prime}(x)=2$ for all $x .$ Must $f(x)=$ $2 x+5$ for all $x ?$ Give reasons for your answer.

In this problem where given a derivative and a point that the function goes through, we're supposed to find the function that goes through it. So the first thing we notice is that F prime of X equals zero. So that means it has a slope of zero everywhere, right? So f as constant slope of zero. And we ask ourselves, What kind of function has a constant slope and even a constant scope of zero? So constant Slope tells us linear and constant slope of zero tells us horizontal, so FX must be a horizontal line, and it goes through the 0.25 So what is the equation of a horizontal line that goes through the 0.25? Well, if it's horizontal, that means the Y value is always the same. And if it goes through the 0.25 that means the Y value needed to be five. So Y equals five would be the equation of the line. And if you put it in the function notation F of X equals five

And this problem, we're given the function f of X equals X squared minus four X minus five and were asked to find the value of the derivative D F inverse DX at X equals zero, which were given is f of five. So to evaluate this, we're just going toe without doing any, you know, finding the universe or anything. We're just going to use the theorem from the section which tells us that the derivative of F in verse at the Point X equals death of A is just one over the derivative of F at X equals. So in this case, were interested in X equals zero, which is f of five, so X equals F five. So if you want to find the derivative at half of five that we just need to find the derivative of F at X equals five. So write this out one more time. The derivative of F in verse evaluated at the point X equals zero, which is F five is the same as one over the derivative of F at X equals five. Now, the purpose of doing this is that this derivative is often times going to be much easier to calculate. We don't even have to find a formula for f inverse to do this. We have a formula for FX, and so we can immediately turn this into a formula for the dread of that right. This is a nice quadratic polynomial. We could just apply the power rule so D E F D X is to X minus four. So if we plug in X is equal to five, then that's two times five minus four, which is 10 minus for which is six. Now. We can just plug this back into our formula down here. We find that the derivative of F in verse X is equal to zero is one over dft Accent X equals five, which is one divided by six.

This question given a function F thanks ICO Jew explode Jew man is far X minus five and excreted and chew And the cook the question asked to find a day I'm the ev inverse I'm in the ANC's Onda bond. ACSI coaches are echoed I'm a five here and by the three room found on the river to after in first function we get you equal to one over prime Ah, off, uh, five. It means knowing that you find the am Bram from here we get it. Could you do X minus far and from here Doesn't again. One dividing by. We're gonna find your hand in Uganda two times five Manus far you again. He called you one over. Tayman is far good You six on. That's gonna be the answer.

So in this session we have to find the derivative with respect to acts of up of you is equal to you square. So since we're looking for a d e f d X, this is really important because we're not just taking the derivative. We have working there derivative with respect to X. But notice here we have a u, not an X. So that means that we're gonna be using the chain rule to find it. So Dft Axe is going to be equal to to you. We're using the power almost flying by the exponents, attracting one. But then since we don't have a next we over you, we also to multiply by the derivative of our inner function right here. So this issue and so we're going to built flying by you Cry s So this is D E F D X on now, we're told that we're looking for dft acts at X is equal to two. The problem also provide some information about what you have to is and what you prime of to us. So you have to is equal to negative five and you prime of two is also equal tonight. But at five so we can plug those numbers. And because we know that that happens when X is equal to two. So Dia Deaniacs at X is equal to two is equal to Q times negative five and times another negative. And so, if you must know that out, we get two times 25 and that gives us 50. We have two negatives. They make a positive, so Dft X is equal to 50 this is the final answer.


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