Use the properties of inverses to determine whether ƒ and g are i...

Question

Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬4 (x+3), g(x) = 4^x + 3

Accepted Answer

Hello, today we're going to be determining whether the two given functions are inverses of each other. So we are given F of X is equal to the logarithm of base five of X plus 11. And we are given G of X is equal to five to the power of X plus 11. So there's gonna be one condition that we need to meet in order to prove that these are inverses of each other. We're going to take the composite function of both of the given functions. And this is going to say that F of G of X is going to equal to X and G of F of X is going to equal to X. So as long as both of these given composite functions are equal to X, that is going to prove that these are inverses of each other. So let's go ahead and take the composite function of F of G of X. In order to get the composite function of F of G of X, we're going to be taking the function G of X which is five to the power of X plus 11 and plugging it in for every variable in the given F of X function or in other words, the composite function of F of G of X is going to be the logarithm of base five of five to the power of X plus plus 11. And simplifying the insight argument of the logarithm is going to leave us with the logarithm of base five of five to the power of X plus 22. Now, there's nothing further that we can do to simplify this logarithm. But it is clear that this is not equal to X. So that means that the composite function F of G of X is not equal to X. And that is going to break our statement that both of the composite functions are equal to X. And what this means is that the answer is going to be B since the first statement is false, these two functions are not inverses of each other. 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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Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬4 (x+3), g(x) = 4^x + 3