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￬2 x+1, g(x) = 2^x-1

Accepted Answer

Hello. Today we're gonna be determining whether the two given functions are inverses of each other. So what we are given is F of X is equal to the logarithm of base nine of X plus six. And we are given G of X which is equal to nine to the power of X minus six. So in order to prove whether these functions are inverses or not is to show that the composite function F of G of X is equal to X. And at the same time, the composite function of G of F of X also must equal to X. So the first thing we're going to need to do is find the values of both of the given composite functions. And if both of the composite functions are equal to X, that shows that these are going to be inverses of each other. So let's go ahead and start by finding the composite function of F of G of X. In order to get F of G of X, we're going to be taking G of X and plugging it in for every variable in the F of X function. So the composite function for F of G of X is going to be the logarithm of base nine of nine to the power of X minus six plus six. Now, we can go ahead and simplify the insight argument of the logarithm because negative six plus six is just going to cancel each other out. And what we are left with is the logarithm of base nine of nine to the power of X. Now, in order to simplify this, we need to understand one logarithmic property. And that states that if you have the logarithm of base A of A to the power of N, as long as the number inside of the logarithm base matches the base of the exponential value, then this entire logarithm is going to simplify to just give us N or in other words, the logarithm of base A of A to the power of N is going to equal to end here, we have a logarithm with base nine and that matches the exponential base which is nine. So that means that the logarithm of base nine of nine to the power of X is going to simplify to just give us X. And that is going to satisfy the first condition. Now we need to go ahead and find the second composite function. So we're going to find the composite function G of F of X. Now, in order to get G of F of X, we're going to be taking F of X and plugging it in for every variable in G of X doing this is going to give us nine race to the power of the logarithm of base nine of X plus six minus six. Now, this is a bit of a messy function, but there's one exponential value or property that we can use to help us solve this suppose that we have some type of exponential value with the base of A and it's raised to the logarithm pro logarithmic exponent with base a of X. As long as the logarithm's base in the exponent matches the base of the total exponential value. This is going to simplify to just give us X in our composite function in the exponent. The logarithm has a base of nine and the overall exponential value has a base of nine. So that means that this first term nine to the power of the logarithm of base nine of X plus six is just going to simplify to give us X plus six. So we can go ahead and rewrite the composite function as X plus six minus six and simplifying this positive six minus six is just going to cancel each other out. And that is going to leave us with X which satisfies the second condition of the inverse property. So now that we've proven that F of G of X is equal to X and G of F of X is equal to X, this verifies that both of the equations are inverses of each other. 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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Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬2 x+1, g(x) = 2^x-1

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