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Numerade Educator

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Problem 23 Easy Difficulty

Find a formula for the inverse of the function.

$ f(x) = e^{2x - 1} $

Answer

$y=\frac{1}{2}(1+\ln x) \cdot \operatorname{So} f^{-1}(x)=\frac{1}{2}(1+\ln x)$

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Video Transcript

we're going to find the inverse of this function, and the first thing I'd like to do is replace F of X with y So we have y equals e to the two X minus one. Now for the inverse we switch X and y So we get X equals E to the two y minus one. And now we sell for Why? So we're going to need logarithms to undo this exponential function so we can take the natural log of the left and the natural log of the right. We can use the power property to bring the two y minus one down and make it a product instead. So we have natural log of X equals a quantity to why, minus one times a natural law. Gov. And remember, the natural log A B is just one. So we have the natural. Lagerback's equals two y minus one. Let's add one to both sides and we have one plus natural log of X equals two y and then we can divide both sides by 21 plus natural log of X, divided by two equals y. Now let's call that F inverse of X. So f inverse of X is one plus natural log acts over to