If (fg)(x) = 4x²−x−5, find f and g.

Question

Accepted Answer

Hello. Today we're gonna be finding the functions F of X and G of X for the given function. So what we are given is the function F times G of X. And we are told that this function is three X squared plus two X minus eight. Now what this means is the notation of F times G of X can be rewritten as F of X multiplied by G of X. So that means that the function that is given to us is a product of two smaller functions. And if we want to be able to find F FX and G of X individually, then what we're going to need to do is factor the given function and we can go ahead and factor this function by using the trial and error method. So the trial and error method states that we want to write the factors for the first term and for the last term and the factors of the first term which I'm going to call the P factors are going to be the combination of all possible factors for three X squared. Now, three is a prime number, which means that the only combination of factors that we have is three X and X. And the factors for the last term, which I'm going to call the Q factors are all the possible negative and positive factors for negative eight. And those factors are going to be negative eight and one or positive eight and one. And they can also be negative four and two and positive four and negative two. Now, what I'm going to do is I'm going to make two pairs of parentheses. And what's gonna go inside these pairs of parentheses is going to be the leading P factors plus or minus the Q factors. So what we're going to do is we're going to pick a combination of factors to plug into our given parentheses and we're gonna test it to see if we can get the original trinomial. So we're going to take three X and X and let's go ahead and try this with negative four and two. So if I plug this into our parentheses, we get three X minus four times X plus two. And now, in order to check to see if this is the correct factorization, we can go ahead and multiply three X with two and the two middle two terms together. And if the product of these terms add up to give us the middle term two X, then this is going to be the correct factorization. So three X times two is going to give us positive six X and negative four times X is going to give us negative four X and six X minus four X is going to give us two X which matches the middle term for the original function. So that means that the function F times G of X is going to factor to three X minus four times X plus two. And these are going to be the possible functions that are F of X or G of X. 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.

If (fg)(x) = 4x²−x−5, find f and g.

Key Concepts

Function Composition

Factoring Quadratic Expressions

Identifying Function Types

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