In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine ...

Question

In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine the domain for each function. f(x) = 3x - 1, g(x) = x - 5

Accepted Answer

Hello, today we are going to be using the two given functions to find the composite function FG and identify its domain. So what does it mean to find the composite function FG? Well FG is telling us that we want to take the product of the function F of X and G of X together. So the functions given to us are F of X is equal to 71 X minus two. We are also told that G of X is equal to X plus 10. In order to get FG of X, we're going to take the product of these two functions. So we will take the function F of X which is 71 X minus two and multiply it to the function G of X which is X plus 10. Now, what we need to do is we need to simplify the composite function. Notice that FG of X is the product of two groups of terms. So in order to simplify this function, we are going to need to foil the two groups together. So we will go ahead and start by taking the first term of the first group, 71 X and multiplying it to each term in the second group. 71 X multiplied by X will give us 71 X squared and 71 X multiplied by 10 will leave us with 710 X. Now, what we are going to do is we are going to take the second term in the first group which is negative two and multiply it to each term in the second group, negative two multiplied by X will give us negative two X and negative two, multiplied by 10 will leave us with negative 20. Now, what we want to do is we want to combine like terms in the function 710 X can be combined with negative two X, 710 X minus two X will leave us with 708 X. And we can't combine 71 X squared with any other term and negative 20 is our only constant. So that means that FG of X simplifies to give us 71 X squared plus 708 X minus 20. This is going to be the simplest form of the composite function FG of X. Now, what about its domain? Well, notice that FG of X is in the form of a polynomial. One special characteristic of a polynomial is that you can plug in any value of X and you will get out any value for FG of X. Or in other words, it's a 1 to 1 function. That means that any value of X will work with this function, which means the domain is all real numbers or from negative infinity to positive infinity. This is always going to be the domain of a polynomial function. So just to recap, we multiplied the function F and G together to get the composite function FG of X, which came out to be 71 X squared plus 708 X minus 20. And because this function is in the form of a polynomial, the domain is going to be all real numbers from negative infinity to positive infinity. With that being said, the solution to this problem is going to be C So I hope this video helps you in understanding how to approach this type of problem. And I'll go ahead and see you all in the next video.

In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine the domain for each function. f(x) = 3x - 1, g(x) = x - 5

Key Concepts

Function Operations

Domain of a Function

Linear Functions

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