Find the inverse

Question

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).

$f\left(x\right)=e^{2x+6}$

Accepted Answer

Welcome back, everyone. Find the inverse function G inverse of X of G of X equals E to the power of three X minus five. And we are given four answer choices. A G inverse is equal to one third L and X plus five thirds. BG inverse of X equals three L and X plus five CG inverse is equal to 1/5 L and X plus 3/5 in DG inverse of X equals five L and X plus three. So first of all, we're given our function and we're going to rewrite it as Y equals E to the power of three X minus five. And since we're solving for the inverse function, we want to switch X and Y. So instead of Y, we're going to have X and instead of X, we're going to have Y, so X equals each to the power of three Y minus five, our goal is to solve or why in order to solve for why we first of all have to bring down the exponent. And we can see that using the rules of logarithms, we're going to use the natural logarithm. This is the common logarithm. So we're essentially taking the natural log of both sides. We're taking Ln of X on the left hand side and Ln of E to the power of three Y minus five on the right hand side. So we have our equation and now based on the rules of logarithms, we can bring that whole exponent in front of the natural log. So that L and of X would be equal to VY minus five multiplied by Ln of E and Ellen of E. Well, essentially it is equal to one. So we can state that Ln of X is equal to three Y minus five. Now let's identify Y valley, first of all, add five to both sides. So L and X plus five would be equal to three Y and now divide both sides by three. So Y equals one third because we're dividing both sides by three. And then we end up with L and X plus five. We're dividing the whole left-hand side, right. So we have to include parentheses. Now, if we distribute the terms, then Y equals one third, L and X plus five thirds, which essentially corresponds to the answer choice. A thank you for watching.

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).
$f\left(x\right)=e^{2x+6}$

Key Concepts

Inverse Functions

Exponential Functions

Logarithmic Functions

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