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Find a formula for the inverse of the function. …

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Problem 22 Medium Difficulty

Find a formula for the inverse of the function.

$ f(x) = \dfrac{4x - 1}{2x + 3} $

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Video by Heather Zimmers

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Video Transcript

all right, we're going to find the inverse of this function, and the first thing I like to do is just call f of X y. So we have y equals four X minus 1/2 X plus three. Now for the inverse we need to interchange or switch X and y So we have X equals before. Why minus 1/2 y plus three. Now we want to isolate Why So let's multiply both sides by the denominator. Two y plus three and we have x times a quantity two y plus three equals four y minus one. Now we can distribute the X to get rid of the parentheses, and we have two x y plus three X equals four y minus one. Remember, our goal here is to isolate Why, So we need to get the white terms on one side of the equation and the other terms on the other side. So I'm going to subtract four wife from both sides, and I have two x y minus four y, and I'm going to subtract three x from both sides. So this equals negative one minus three x. All right now, it's factor. Why out of both terms on the left and we have why Times a quantity two X minus four equals negative one minus three x. Finally, we can divide both sides by two x minus four. So we have y equals negative one minus three x over two X minus four. Now it looks a little bit better with fewer negative signs if we multiply the top by negative one and the bottom by negative one. So that would be one plus three x over four minus two x It's equivalent. It just looks a little more slick. And finally, let's go ahead and rename this instead of calling it Why, let's call it F inverse of X. Make a little more space here F and bursts of X. There we go.

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