Find f(g(x)) and g (f(x)) and determine whether each pair of f...

Question

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=3x+8 and g(x) = (x-8)/3

Accepted Answer

Hello. Today we're gonna be verifying that the given functions of F of X and G of X are inverses of each other. So the F of X function given to us is four over X minus four. And the G of X function given to us is four over the quantity of X plus four. So in order to verify that both of these functions are inverses of each other, we're going to be taking two composite functions. We're going to take the composite function F of G of X and we're going to take the composite function G of F of X. And what we're looking for is that both of these composite functions give us an output of X. If both composite functions are equal to X, then this verifies that they are inverses of each other. And if one of them does not equal to X, then that verifies that they are not inverses of each other. So let's take the first composite function F of G of X. In order to get this composite function, we're going to be taking the function G of X and plugging it in for every variable in F of X. And that is going to change the function of F of X to four over the quantity four over X plus four minus the quantity of four. So how do we simplify this complex fraction? Well, we can rewrite the numerator of four as a fraction 4/1. And if we wanted to simplify the denominator to give us one, we'll multiply the denominator by its reciprocal X plus 4/4. And if we multiply the denominator by the reciprocal, we must also multiply the numerator by the reciprocal X plus 4/4. What this is going to do is that this is going to eliminate the denominator and leave us with one. And now we just need to multiply the numerators together. So simplifying this is going to leave us with 4/1, multiplied by X plus 4/4 minus the quantity of four using cross elimination, we can eliminate the four in the numerator and denominator and reduce them to one and X plus four times one is going to give us X plus four. So the composite function is going to simplify to X plus four minus four and combining the constants together four minus four is going to cancel each other out leaving us with just X. So we've just verified that the composite function F of G of X is equal to X. Now, we need to find the second composite function G of F of X and just like before. In order to take this composite function, we're going to take the F of X function and plug it into every variable in the G of X function doing this is going to change the G of X function to give us four over the quantity of four over X minus four plus four. Now, in the denominator, if we've added negative four and four together, that's going to cancel each other out, leaving us with 4/4 over X. Once again, we have a complex fraction, we can rewrite the numerator as 4/1. And if we want, if we want to reduce the denominator to just one, we're gonna multiply the denominator by its reciprocal X over four. And we'll be doing the same to the numerator multiplying it by X over four. The denominator is gonna reduce to just give us one. And what we are left with is 4/1, multiplied by X over four. And using cross reduction again, we can simplify the four in the numerator and denominator to leave us with one and one times X is just going to give us X. So this verifies that the composite function G of F of X is equal to X. And since both of the composite functions are equal to X, this verifies that F of X and G of X are inverses of each other. And 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.

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=3x+8 and g(x) = (x-8)/3

Key Concepts

Function Composition

Inverse Functions

Algebraic Manipulation

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