Find the inverse

Question

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

$f\left(x\right)=\ln\left(3x+1\right)$

Accepted Answer

Welcome back, everyone. Find the inverse function G inverse of X of G of X equals two LNX minus three plus four. And we're given four answer choices a four E to the power of X divided by three plus three, B three, E to the power of X divided by two plus four, C eats the power of X minus three, divided by two plus four and de to the power of X minus four divided by two plus three. First of all, our function G of X is equal to two LNX minus three plus four. And we are going to replace G FX with Y. So that's our Y coordinate, which is two LNX minus three plus four. 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 and our goal is to solve or why? Which is going to be our inverse function. So first of all, let's isolate the natural log, we're going to subtract four from both sides, meaning X minus four is equal to two LM of Y minus three. Now, we have to divide both sides by two to get rid of the leading coefficient. Therefore, X minus four, divided by two would be equal to Ln of Y minus three. And now from here, we want to get rid of that log, right? And we have to recall that whenever we have a, the power of a log of base A of number B, this is equal to B and this allows us to get rid of the log. So we can take the anti log of both sides. If we take E which is base of Allen, then eats the power of left hand side eats the power of X minus four, divided by two will be equal to eat the power of Ln which is log E of Y minus three. And we can see that on the right hand side, we have A to the power of log A B which essentially follows the formula. Therefore, the left hand side eats the power of X minus four divided by two is equal to Y minus three, right based on the property. And now to, so for Y, we have to add three to both hand sides. So Y equals, it's the power of X minus four divided by two plus three. Now, if we look at the answer choices, we can conclude that the correct answer to this problem would be option D and that's it. Thank you for watching.

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).
$f\left(x\right)=\ln\left(3x+1\right)$

Key Concepts

Inverse Functions

Natural Logarithm

Domain and Range

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