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) = 5^x + 1

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 14 rate to the power of X plus eight. Now, the first thing we want to do whenever we're trying to find an inverse is to replace F of X with Y. So we're going to rewrite the given equation as Y is equal to the race of the power of X plus eight. Then the next step is to rep replace Y with X and X with Y or in other words, we're going to swap the decisions of X and Y. So that allows us to rewrite the equation as X is equal to 14 rate of the power of Y plus eight. Now, what we want to go ahead and do is solve for Y in order to get our inverse function. So the first thing we can go ahead and do is subtract eight to both sides of the equation that is going to give us X minus eight is equal to 14 raised to the power of Y. Now, we need to figure out a way to bring Y down from the exponential position. While we have a logarithmic property that is going to help us do that. The logarithmic property states that if you have the log base of a, of a race to the power of N, as long as the number in the base of the logarithm matches the base of the exponential value inside the logarithm. Then all of this is going to cancel out to just leave us with the exponent or in other words, the log base A of A to the power of N is just going to leave us with N. So the first thing we're gonna want to do is we're going to take a logarithm with a base to both sides of the equation. But what base should we use? Well, we have this exponential value with a base of 14. So why not try log base of 14? If we take the log base of 14 to both sides of the equation, we can rewrite this as the log base of 14 of X minus eight is equal to log base of of 14 raised to the power of Y. Now notice that we can go ahead and apply our logarithmic property to the right hand side of the equation. And the reason for that is because we have a log base of 14 and we have an exponential value with the base of 14. All of this is going to cancel out and just leave us with Y. So what we are left with is Y is equal to log base of 14 of X minus eight. 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 equal to log base 14 of X minus eight. And this is going to be the inverse function of the given 1 to 1 function. And with that being said, the answer to this problem is going to be D. 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) = 5^x + 1

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