Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See...

Question

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒg)(4)

Accepted Answer

Hello, today we are going to be using the two given functions to evaluate F times G of negative one. So the problem tells us that the function F of X is equal to three X cubed plus one. We are also told that the function G of X is equal to negative seven X plus six. And we want to use both of these functions to find the value of F times G of negative one. So how can we start this process? Well, F times G of negative one is telling us that we first need to take the product of the function F and G so F times G of X afterwards, once we have this function, we're going to plug in the value of negative one. And that's going to be the value of F times G of negative one. So first, let's go ahead and evaluate what the function F times G of X is going to be. This is telling us to take the product of F of X and G of X. So that is going to be F of X which is three X cubed plus one multiplied by the function G of X which is negative seven X plus six. When we place these two functions into F times G of X, notice that we have the product of two factors of X. In order to simplify this, we're going to need to foil the two groups together to start that process. We are going to take the first term of the first group which is three X cubed and multiply it to each term in the second group, starting with negative seven X three X cubed multiplied by negative seven X will give us the value of negative 21 X to the fourth power. Next, we'll take three X cubed and multiply to positive six, which is going to give us positive 18 X cubed. Now we'll take the second term of the first group which is positive one and multiply it to each term in the second group, one times negative seven X will give us negative seven X and one multiplied by six will give us positive six. We cannot combine any of the terms or simplified further. So F times G of X is going to equal to negative 21 X to the fourth plus 18 X cubed minus seven X plus six. Now that we have the function F times G of X, we're going to want to plug in the value of negative one in order to evaluate F times G of negative one. So in order to find the value of F times G of negative one, we're going to take the value of negative one, plug it into the function. And that is going to allow us to replace every X in the function with the value of negative one. That is going to give us F times G of negative one is equal to negative 21 multiplied by negative one to the fourth power plus 18 multiplied by negative one to the third power minus seven, multiplied by negative one plus six. Now, what we need to do is we need to algebraically simplify whenever you have negative one raised to an even power such as four, we are going to get positive one as the result. So negative one raised to the fourth power is going to give us positive one. So we have negative 21 multiplied by one next. Anytime we have negative one race to an odd power such as three, the result is going to be negative one. So we will now have 18 multiplied by negative one, negative seven, multiplied by negative one is going to give us positive seven. So we have plus seven and finally, we add the constant at the end of the function which is positive six, negative 21 multiplied by one will just give us negative 21 and positive 18 multiplied by negative one will give us negative 18. So we are left with negative 21 minus 18 plus seven plus six. And once we take the differences and sums of all the terms we are going to be left with negative 26. So what this means is that the value of F times G of negative one is going to equal to negative 26. And that tells us that the answer to this problem is going to be B So I hope this video was helpful in helping you understand how to approach this process. And I'll go ahead and see you all in the next video.

Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒg)(4)

Key Concepts

Function Composition

Evaluating Functions

Quadratic Functions

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