find-an-explicit-formula-for-f-1-and-use-it-to-graph-f-1-f-and-the-line-yx-on-the-same-screento-chec

Question

Find an explicit formula for $f^{-1}$ and use it to graph $f^{-1}$ , $f,$ and the line $y=x$ on the same screen.To check your work, see whether the graphs of $f$ and $f^{-1}$ are reflections about the line.
$$f(x)=x^{4}+1, \quad x \geqslant 0$$

Amrita Bhasin · Accepted Answer

this question asked us to find an explicit formula for the inverse function. First things first, swap variables and soul for Why take the fourth route? We now have a graph that looks like this. You can see f an inverse function of off our reflections about this line. The line and black is white.

Heather Zimmers · Answer

all right. In this problem, we have f of X, and what we want to do is find its inverse. So let&#x27;s go ahead and change f of x toe. Why? So we have y equals one plus each of the negative X and then to find the inverse, we want to switch positions of X and y. So now we have X equals one plus each of the negative. Why? And now we&#x27;re going to sell for why So let&#x27;s start by subtracting one from both sides, and then we&#x27;ll take the natural log of both sides that will leave negative wide by itself. And then we&#x27;ll multiply both sides by negative one to get why by itself. And then instead of calling it why we can call it f inverse of X. So what we&#x27;re going to do is graph this graph the original f of X and graph the line y equals X and look and see if the two functions are reflections across the line Y equals X. So we grab a calculator, press y equals. We have our original function f of X here, and we have our inverse that we just found and we have y equals X, and I&#x27;m going to look at the&#x27;s in a zoom decimal window so that zoom number four. So the question is, do they look like reflections across the line? Y equals X? Definitely they dio.

Carlos Pinilla · Answer

um So we&#x27;re where&#x27;d you get the following function? But Ziegel toe, go win us the X. They want to compute, but his fingers. Well, if this is you want the soul for you. Call these. Why is, uh, to minus into the ex I want to solve for X, so thanks. So, um, we gonna rex so Well, you can, uh, what we&#x27;re going to hear is you got with that wonder wearing that undid its obstruct. Why there? So you did. E x would be equal to do. Minus what? Right? That&#x27;s a little Dana. You can&#x27;t think did not tolerate movies expression. So what did not relate more on both sides? Well, naturally, have you seen yourself expansion? So besides, you get eggs, That&#x27;s I get not from Oregon. Two my nose. What? So this is saying that Well, if Congress why is your ex but us of what is this function? Um, so that is there, Inglis? So, uh, or we we&#x27;re going to draw this ocean. You can do all these money is the red one. They&#x27;re lazy. Regional functionaries. D&#x27;oh Minus. Did he x Well, as you can see his number. This part here It&#x27;s always negative. So it was gonna be there. Red Gruff is not Is never hired onto. So all these blue one is there is the response shin. So return in terms of, uh, effects is really look too Miners X. So will you. You drill your You&#x27;re lying is the line. Believe me, that&#x27;s a line if you throw this line, huh? She goes. Why find the game was graphically is like reflecting is acceptable. You can see you this right point. It was still that blue point. Uh, the point is, red points. There&#x27;s the same because he was in the line. These points boats over there. Look at that point. These red point. No, uh, over there. That&#x27;s fine. So So it is because he&#x27;s just a reflection of all perfect. We&#x27;ll reflect these. They&#x27;re autographed. He&#x27;s lying. I see. When I get big rule, like so these one. We&#x27;re going, Uh, well, this has a nice sheen. Shimmer touching the line, too. But he&#x27;s, uh, horizontal visited. And then for the structure for the elevator. We&#x27;re neighbors. Battery. Some ability become some vertical, isn&#x27;t it? Why equal? So buy their affection process. I mean, well, it&#x27;s not defined on X equals. So but only for all right. It is more than no, as you can see.

Amy Jiang · Answer

Okay, so we&#x27;re asked to solve the following functions Were inverse function so cyber rewriting it as wise? It was two months. Feature X. Now, um, it&#x27;s in to change our X and Roy. And now it&#x27;s all for wide. So what started price of tracking one on both sides for sorry to on both sides, multiplying by negative, honorable, suddenly getting of the X and then plus two is equal to use it. Why? And now let&#x27;s take the natural log of both sides so we can bring down or power. Do we get Ellen of two? Money is equal to what and that&#x27;s equal to our interest function.

Heather Zimmers · Answer

all right to start this problem, we have function f of X, and we&#x27;re going to find its inverse. And I think it will be convenient if we call it why, instead of f of X, So to find the inverse, the first thing we want to do is switch X and y. So now we have X equals the square root of four y plus three and then we want to solve for y. So we&#x27;re going to square both sides, and then we&#x27;re going to subtract three from both sides, and then we&#x27;ll divide by four. Now, let&#x27;s call this inverse f inverse instead of calling it. Why so f inverse of X equals X squared minus 3/4. Okay, so now we grab a calculator and we want to graph what we just found. F enders X squared minus 3/4, as well as the original function F for X plus three as well as the line y equals X. And I&#x27;m going to use a decimal window zoom decimal, which has a nice spread of equal spacing for X and y axes. Okay, so here&#x27;s what we&#x27;re looking at. So if you pressed Trace you can tell which graph is which. Now the blue one is the inverse. That was the parabola. Now we need to ignore part of it. We don&#x27;t actually want the full parabola because the square root function was not a full proble. It was only half so we only want the right side. Basically. So what we could Dio is go back to our graph and imagine that we&#x27;re just getting rid of this extra piece here. We don&#x27;t want any of that. It goes to Oh, roughly here. Okay, that&#x27;s him out. We want now as we look at it. What&#x27;s the relationship that we notice? Do we see that these graphs are reflections about the line Y equals X? They definitely look like it.

Katelyn Chen · Answer

It&#x27;s question. We have redrawn autograph vaccine Teoh to the X and we asked the graph after Brexit see constituted ex post one in the past one, causing our entire graft district over to off by one unit. And so this is what a graph looks like and this is our answer.

Problem

Find an explicit formula for $f^{-1}$ and use it …

Problem 27 Medium Difficulty

Find an explicit formula for $f^{-1}$ and use it to graph $f^{-1}$ , $f,$ and the line $y=x$ on the same screen.To check your work, see whether the graphs of $f$ and $f^{-1}$ are reflections about the line.
$$f(x)=x^{4}+1, \quad x \geqslant 0$$

Answer

$f^{-1}(x)=\sqrt[4]{x-1}$

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James Stewart

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$f^{-1}(x)=\sqrt[4]{x-1}$

Biocalculus Calculus for the Life Sciences

Catherine R.

Kayleah T.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Catherine R.

Kayleah T.

Caleb E.

Michael J.

James Stewart

Biocalculus Calculus for the Life Sciences