Write an equation for the inverse function of each one-to-one fun...

Question

Write an equation for the inverse function of each one-to-one function given. ƒ(x) = (1/3)^x

Accepted Answer

Hello, today we're gonna be finding the inverse function of the given 1 to 1 function. So what we are given is F of X is equal to 3/17, raised to the power of X. Now, the first thing we're gonna want to do in order to find the inverse is to replace F of X with Y. So we're going to rewrite the equation as Y is equal to 3/17 race to the power of X. Then the next step is to swap the positions of X and Y. So we can rewrite the equation as X is equal to 3/17 race to the power of Y. Now, what we need to do is to go ahead and solve for why? Well, we're going to need to figure out a way to bring Y down from the exponential position and we have a logarithmic property that is going to help us do that. So suppose that we have the logarithm of a, of a race to the power of N as long as the number in the logarithmic base matches the base of the exponential value. This is going to simplify to just leave us with N or in other words, the log base A of A to the power of N is going to simplify to give us N. So we're going to have to take a logarithm of both sides of the equation. But on the right hand side of the equation, we have an exponential value with the base of 3/17. So let's go ahead and try log of base 3/17 to both sides of the equation. That is going to give us the logarithm of base 3/17 of X is equal to the logarithm of base three. Over 17 of 3/17 raced to the power of Y in this given logarithm, the base of the logarithm and the base of the exponential value are the same number. So this is going to simplify to just leave us with Y when we apply our logarithmic property to the right hand side. So we can go ahead and rewrite the equation as Y is equal to the logarithm of base 3/17 of X. And finally, we're going to replace Y with F inverse of X. So the inverse function is going to be F inverse of X is equal to log base 3/17 of X. And this is going to be the final form of the inverse function. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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Write an equation for the inverse function of each one-to-one function given. ƒ(x) = (1/3)^x

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