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Numerade Educator

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Problem 57 Hard Difficulty

(a) We must have $e^{x}-3>0 \Leftrightarrow e^{x}>3 \Leftrightarrow x>\ln 3$. Thus, the domain of $f(x)=\ln \left(e^{x}-3\right)$ is $(\ln 3, \infty)$.
(b) $y=\ln \left(e^{x}-3\right) \Rightarrow e^{y}=e^{x}-3 \Rightarrow e^{x}=e^{y}+3 \Rightarrow x=\ln \left(e^{y}+3\right),$ so $f^{-1}(x)=\ln \left(e^{x}+3\right)$.
Now $e^{x}+3>0 \Rightarrow e^{x}>-3,$ which is true for any real $x,$ so the domain of $f^{-1}$ is $\mathbb{R}$.

Answer

a) $(\ln 3,+\infty)$
b) $f^{-1}(x)=\ln \left(e^{x}+3\right) ;$ Domain: $(-\infty, \infty)$

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Video Transcript

okay. We want to find the domain of F of X. And we know for the standard natural log function y equals natural log of X. That the domain is X is greater than zero. What you take the natural log of has to be a positive. Okay, so that means that in our problem, e to the X minus three has to be greater than zero. So we have e to the X minus three is greater than zero. We can add three to both sides. Be to the X is greater than three. Then we can take the natural log of both sides. Natural log of each of the X is greater than natural Log of three. A natural log of each of the exes. Just x The natural log function cancels with the E to the X function, leaving you with just X so X is greater than natural log of three. So that's the domain. We can write that as national log of three to infinity. All right, now let's find the inverse function. So instead of calling the original function f of X, I'm going to call it why Why equals natural log of each of the X minus three and then for the inverse. We're going to switch X and y so we get X equals natural log of each of the Y, minus three. Now we want to rearrange into exponential form. Remember, Natural log means Long Basie. So we have e to the power X equals E to the Y, minus three. Now we want to solve this for way. So let's add three to both sides. E to the X Plus three equals each of the wide. And then let's take the natural log of both sides and we get natural log of each of the X plus three equals. Why? So there's are inverse function instead of calling it why we can call it f inverse of X. How about its domain? Well, just like the previous problem where we saw that e to the X minus three had to be greater than zero. In this case, E to the X Plus three has to be greater than zero. But we know that E to the X is always greater than zero. So something greater than zero plus three must be greater than zero. So the domain is going to be all real numbers