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) = 5^x, g(x) = log￬5 x

Accepted Answer

Hello, today we're going to be determining whether the two given functions are inverses of each other. So what we are given is F of X is equal to A, to the power of X and we are given G of X is equal to the logarithm of base eight of X. So in order to prove whether these functions are inverses of each other, we need to prove one property. And that states that the composite function of F of G of X is equal to X. And at the same time, the composite function G of F of X is going to equal to X. As long as this statement is true, this is going to prove that the two given functions are inverses of each other. So let's go ahead and start by finding the composite function F of G of X. In order to get this composite function, we're going to take the function G of X which is the logarithm of base eight of X and replace every variable in F of X with this function. So doing this is going to give us eight race to the power of the logarithm of base eight of X. Now, in order to simplify this, there's one exponential property we need to understand, suppose we have an exponential value with the base of A and its race to the power of the logarithm of base A of B. As long as the base in the exponent matches the base of the entire exponential value. This is going to simplify to just leave us the insight argument of the logarithm and the exponential value or in other words, a race, the power of the log base A of B is going to simplify to just give us B here in this composite function, the overall exponential value has a base of eight. And in the exponent, the logarithm has a base of eight because these two values match each other. This is going to simplify to just X which proves the first condition of the inverse property. Now, what we need to do is we need to find the other composite function which is G of F of X. So now we're gonna find G of F of X. And in order to do this, we're going to take F of X and plug it in for every variable in the given G of X function. And that is going to give us the logarithm of base eight of eight to the power of X. Now, in order to simplify this, there's one logarithmic property we need to understand, suppose we have the logarithm of base A of A to the power of N. As long as the number inside the logarithmic base matches the base of the exponential value. This entire logarithm is going to simplify and leave us with just N or in other words, the logarithm of base A of A to the power of N is going to equal to N. So in our given composite function, the logarithm has a base of eight and the exponential value inside the logarithm also has a base of eight. So this is going to simplify to just leave us with the exponent. And that means that the composite function of G F F of X is going to equal to X. And that proves the second part of the inverse property. So we've shown that F of G of X is equal to X and G of F of X is equal to X. So that proves that yes, both of these functions 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 approve 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) = 5^x, g(x) = log￬5 x

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