Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^(x-5)

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Hey, everyone here we are asked to determine the inverse of F of X is equal to E to the power of two X minus 15. 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 15, all divided by two or the natural log of X minus 15, all divided by two. And there are two possibilities for the domain and range one is from zero to positive infinity and the other is from negative infinity to positive infinity. So to begin determining the inverse of our function, we first want to simply let F of X be equal to Y. So now rewriting our function, we have Y is equal to E to the power of two X minus 15. So now we just want to manipulate this equation to solve for X. So here we see that we have the X term as the exponent of E. And so to get this X term to be outside of the exponent, we first need to simply take the natural log of both sides. So taking the natural log, we have the natural log of Y is equal to the natural log of E raised to the power of two X minus 15. So now to get our X term outside of the exponent, we need to recall the logarithmic property which states that the natural log of A raised to the power of X is equivalent to X times the natural log of A. So now rewriting the right side of our equation, we have the natural log of Y is equal to the quantity of two X minus 15 times the natural log of E. So now we just need to recall that the natural log of E is the same as the natural log of or the log base E of E. And the log base E of E is simply equivalent to one. So we can simplify the right side of our equation by having the natural log of E simplified to one. So now rewriting our equation, we have two X minus 15 is equal to the natural log of Y. So now we can begin simply solving for X. So first, we just want to add 15 to both sides of the equation. And so we have two X is equal to the natural log of Y plus 15. Now to get rid of the coefficient of X, we divide both sides by two and we get X is equal to the natural log of Y. Plus 15, all divided by two. And so now the last step in determining our inverse is to have X be equal to F inverse of X and have Y be equal to X. So now replacing these terms in our equation, we are left with F inverse of X is equal to the natural log of X plus 15 all divided by two. So now we have found our inverse of our given function. So we just need to now determine its domain and range. So beginning with the domain here we know that all values within the natural log must be greater than zero. So here we know that X must be greater than zero since X is within the natural log. So this tells us that our domain is going to be from zero to positive infinity. And now for the range, we just need to recall that the value of the natural log of X varies from negative infinity to positive infinity. So this tells us that our Y values our output or our range is going to be from negative infinity to positive infinity. So here we have found our inverse function as well as determined its domain and range. And so we are left with answer choice. A F inverse of X is equal to the natural log of X plus 15, all divided by two. The domain is from zero 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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Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^(x-5)

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