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) = 2 ln 3x

Accepted Answer

Hey, everyone here we are asked to determine the inverse of the function F of X is equal to 17 times the natural log of 12 X. 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 inverse function is either positive or negative 1, 12 times E raised to the power of X divided by 17. And the domain is either from zero to positive infinity or from a negative infinity to positive infinity. And the range also has two possibilities either from negative infinity to zero or from zero to positive infinity. So our first step in determining the inverse is to just have F of X be swapped with or be equal to Y. So now just rewriting our equation, our function we have Y is equal to 17 times the natural log of 12 X. So now we just want to manipulate this equation to solve for X. So our first step is to just move over this coefficient of the natural log to the other side of the. And we do this by simply dividing both sides by the coefficient which is 17. So now we just have Y divided by 17 is equal to the natural log of 12 X. So now the next step is to get our X term to be outside of the natural log. And so to do this, we have to let E be raised to the power of the right side and the left side of the equation. So doing this, the left side of the equation becomes E raised to the power of Y divided by 17. And the right side of the equation becomes E raised to the power of the natural log of 12 X. So now we can see here that E and the natural log on the right side of the equation cancel each other out. And so we are left with 12 X is equal to E raised to the power of Y divided by 17. So now we just want to get X completely alone by dividing both sides by 12. And so we are left with X is equal to 1 12 times E raised to the power of Y divided by 17. And so now the last step in determining the inverse is to just have X be equal to F inverse of X and then have Y be equal to X. So now just replacing these terms, our equation or inverse function becomes F inverse of X is equal to 1 12 times E raised to the power of X divided by 17. So here we have determined the inverse function of our given function. So all that is left is to determine its domain and range. So if we recall the domain are all X values such that our function is defined. And here we see that X is in the exponent and numerator of our E term here. And so we can see that we can input any X value into our function and it will still be defined. Therefore, our domain is from negative infinity to positive infinity. Now, looking at our range, we have to think of the Y values the outputs of our function such that our function is defined. And if we recall the range of an exponential function starts from zero and goes up to infinity. Therefore, our range is from zero to positive infinity. So here we have found our domain and range of our inverse function. And so we are left with answer choice D where F inverse of X is equal to 1 12 times E raised to the power of X divided by 17. The domain is from negative infinity to positive infinity. And the range is from zero 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) = 2 ln 3x

Key Concepts

Inverse Functions

Properties of Logarithmic Functions

Domain and Range of Functions

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Find ƒ-1(x), and give the domain and range. ƒ(x) = 2 ln 3x

Key Concepts

Inverse Functions

Properties of Logarithmic Functions

Domain and Range of Functions

Watch next

Find ƒ^-1(x), and give the domain and range. ƒ(x) = 2 ln 3x