In Exercises 31–50, find fg and determine the domain for each fun...

Question

In Exercises 31–50, find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

Accepted Answer

Hello. Everyone in this video. We're going to be looking at this practice problem where we want to find F divided by G. And write the domain of the new composite function F divided by G. We are given that F of X is equal to nine X plus two divided by X squared minus 49 G of X is equal to eight X plus nine divided by x squared minus 49. So our first step is going to be to find the composite function F F times G. So F times G is the same as F of X times G of X. And we're given F of X and G of X. So I can go ahead and plug those values in. So I have F. Of X which is nine X plus two divided by X squared minus 49. And it's being multiplied by G. F X which is eight x plus nine divided by X squared minus 49. So if I complete this multiplication, my numerator becomes nine X plus two times eight X plus nine divided by X squared minus 49 squared. So this becomes my composite function, F Times Street. And now we need to write the domain of this new composite function. And in this case because we're working with a fraction, the only place where the X value would give us an invalid or undefined answer would be if the denominator is equal to zero. So remember that for the domain we're trying to find the range of X values that will ensure a defined answer. So we just want to make sure that our denominator Does not equal zero. So our denominator is x squared minus it cannot equal zero. So if I solve for X, that's a sorry the whole expression is squared but if I do the square root of that I have x squared - cannot equal and it's still zero because the square root of zero is still zero. So now I can solve for X And I'm gonna add 49 to both sides. So that leaves me with X squared cannot equal positive 49. And if I do the square root of both sides, I finally have X by itself and I have the X cannot equal plus or minus seven. Notice that I did plus or minus seven because the square root of 49 could be positive seven or negative seven. So now I have this expression for X values that do not work. So let's build an interval around that. So I know that X can go from negative infinity two negative seven And it can't equal -7. So I'll go ahead and do a parentheses But it can go from -7 to positive seven. And again I'll do a parenthesis after the positive seven because X cannot equal seven. Exactly. And I'll do a final union because X can go from seven to positive infinity. So this becomes my interval for X values that give me a defined answer. And in this case we want to make sure that we integrate the previous functions domains into our new composite function. So if we look at the previous functions, you can see that they both have the same denominator X squared minus 49. So we want to make sure that we include the range of X values that are valid in their domain. So we want to make sure that if any values of X were excluded in the previous functions, they're included or they're also excluded in the new composite function. Because the composite function is built off of those old functions. So if we follow the same procedure, I want to make sure that X squared -49 does not equal zero. And if we do that, I'll add 49 to both sides. And I'm left with X squared cannot equal positive 49. And again I'll do the square root Leaving me with X cannot equal plus or -7. And you can see that both the old functions and the new functions have the same interval. So I don't have to worry about changing my domain. And this becomes my final answer. So my domain is negative infinity to negative seven. Union with negative seven to positive seven. Union with seven to positive infinity. And f times G is equal to nine X plus two times eight X plus nine divided by X squared minus squared. And you can see that this aligns with answer choice C so hopefully this video helped and see you next time.

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In Exercises 31–50, find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)