Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^(x+1) - 4

Question

Accepted Answer

Hey, everyone here we are asked to determine the inverse of F of X is equal to E to the power of three X minus five plus seven. And we are also asked to provide the domain and range of the inverse. Here we have four answer choice options. A through T each with the inverse function being either the natural log of X minus seven plus five all divided by three or the natural log of X plus seven plus five all divided by three. And now for the domain and range, there are three of these same possibilities. One is from positive seven to positive infinity or from negative seven to positive infinity or the third possibility from negative infinity to positive infinity. So now to determine the inverse of our function, we first want to let F of X be equal to Y. So now just rewriting our function, we have Y is equal to E to the power of three X minus five plus seven. So now we just want to manipulate this equation to solve for X. So the first step is to just move over this positive seven to the left side of our equation. And we do this by simply subtracting seven from both sides. So we have Y minus seven is equal to E to the power of three X minus five. So now we want to get this X term to be outside of the exponent for E. So to do this, we need to just take the natural log of both sides of the equation. So taking the natural log, we have the natural log of Y minus seven is equal to the natural log of E raised to the power of three X minus five. Now to simplify the right side of our equation, 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 you utilizing this formula to simplify the right side of the equation, we get the natural log of Y minus seven is equal to the quantity of three X minus five times the natural log of E. Now, we just need to call that the natural log of E is the same as log base E of E. And this expression is simply equivalent to one. So now we can simplify the right side of our equation by simplifying the natural log of E to be equal to one. So now rewriting our equation, we have three X minus five is equal to the natural log of Y minus seven. So now we want to work on getting X alone on the left side of the equation. So first, we just need to add five to both sides. And so we get three, X is equal to the natural log of Y minus seven plus five. Now to get rid of the coefficient of X, we just divide both sides by three and we get X is equal to the quantity of the natural log of Y minus seven plus five all divided by three. And so now the last step in writing or determining our inverse is to just have X be equal to F inverse of X and have Y be equal to X. So now just replacing our terms, we have the function F inverse of X is equal to the natural log of X minus seven plus five all divided by three. And so now we have found our inverse for our given function. So all that is left to do is to determine its domain and range. So beginning with the domain, if we recall the domain is all of the X values for which our function is defined. And here we see that our X term is within the natural log and we know that values within the natural log must be greater than zero. So we can take the expression that is within the natural log which is X minus seven and set it to be greater than zero. Now, solving for X, we just add seven to both sides. And we get that X be greater than seven. So now we just need to write this inequality as a domain in interval notation. And so our domain will be from seven to positive infinity. And now for our range, we simply have to recall that the value of the natural log of X varies from negative infinity to positive infinity. So since our outputs vary from negative infinity to positive infinity, our range is also from negative infinity to positive infinity. So here we have also determined our domain and range for our inverse function. And so we are left here with answer choice A where F inverse of X is equal to the quantity of the natural log of X minus seven plus five divided by three. The domain is from seven 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.

Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^(x+1) - 4

Key Concepts

Inverse Functions

Exponential Functions

Domain and Range

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