Find ƒ^-1(x), and give the domain and range. ƒ(x) = ...

Question

Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^x + 10

Accepted Answer

Hey, everyone here, we are asked to determine the inverse of F of X is equal to E to the power of X plus 53. And we are also asked to provide the domain and range of the inverse. Here we have four answer choice options. A through D each with the inverse function being either the natural log of X plus 53 or the natural log of X minus 53. And now for the domain and range, there are the same three possibilities. One is from 53 to positive infinity or from negative in negative 53 to positive infinity or negative infinity to positive infinity. Now, to begin determining the inverse of our function, we first just want to let F of X be equal to Y. Now rewriting our equation, we have Y is equal to E to the power of X plus 53. So now we just want to manipulate this equation to solve for X. So the first step is to just move over this positive 53 to the left side of the equation. And we do this by simply subtracting 53 from both sides. And so we have E to the power of X is equal to Y minus 53. And so now we just want to get X to be outside of this exponent of E. And so we first need to take the natural log of both sides. So doing this, we have the natural log of E raised to the power of X is equal to the natural log of Y minus 53. And so now we need to recall the logarithmic property where the natural log of A raised to the power of X is equivalent to X times the natural log of A. And now we will use this property to simplify the left side of our equation. And so we will have X times the natural log of E is equal to the natural log of Y minus 53. And so the next step is to recall that the natural log of E is the same as log based E of E and this is simply equivalent to one. So here we can simplify the left side of the equation by having the natural log of E be simplified to just one. So doing this, our equation becomes X is equal to the natural log of Y minus 53. So now the last step in determining our inverse is to have X be equal to the F inverse of X and Y is just going to be equal to X. So now replacing these terms, we get the inverse function F inverse of X is equal to the natural log of X minus 53. So now we have determined the inverse of our given function. So all that is left is to determine its domain and range. So beginning with the domain, we see that our X term is within the natural log. So we have the expression X minus 53 within the natural log. And we know that any term within the natural log must be greater than zero. So we can set X minus 53 greater than zero. Now, we can solve for, for X within this inequality. So we just need to add 53 to both sides. And we get that X must be greater than 53. So now writing this inequality as a domain in interval notation, we see that our domain is from positive 53 to positive infinity. And now for our range, if we think of the possible outputs of our function here, we will recall here that the value of the natural log of X varies from negative infinity to positive infinity. Therefore, our range is from negative infinity to positive infinity. So here we have determined the domain and range of our inverse function. And so we are left with answer choice A where F inverse of X is equal to the natural log of X minus 53. The domain is from 53 to positive infinity. And the range is from negative infinity to positive infinity. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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