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)= = 8x/(x - 2), g(x) = 6/(x+3)

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to find F times G. And write the domain of F times G. And we're given that F of X is equal to 27 X divided by X plus nine and G of X is equal to nine divided by x -7. So in this case our first step is going to find is going to be to find F times G and F times she is the same thing as X times G of X. So we have the F function F of X and the G function G of X. And we're given F of X and G of X. So we can write those in. So F times G is equal to F of X which is 27 X divided by X plus nine times G of X which is nine divided by X -7. And in this case you can see that we can complete this multiplication and if we do that we have x times nine which is 243 X divided by X plus nine times x minus seven. So you can see we've completed the multiplication and we're left with this equation. So now we need to find the domain of the equation, recall that the domain is the range of X values. That will ensure our equation is in the real remote oil, that it will give us a real number. So we want to make sure that F of G which is equal to 243 X divided by X plus nine Times X -7 will give us a real number. And the way to ensure that is to make sure that we have all real numbers as a possibility when we plug in X. So you can see here that When we plug in negative 100 to this top x we'll get a negative number but it will still be real. So we have to worry about the denominator here with these exes because recall that if the denominator of a fraction is zero, that answer is not defined. So we want to make sure that that doesn't happen. So we want to make sure that X-plus nine Does not equal zero Or X -7 does not equal zero. And in this case we want both of them to not be able to equal zero because if even one of them is equal to zero, the denominator will become zero because any number times zero will be zero. So we want to make sure to find the range of X values that will ensure that both of them will be not equal to zero. So if we look at this first equation X plus nine cannot equal zero. If I subtract nine to isolate X. I get that X cannot equal negative nine. If I do the same thing to my second equation, I'll add seven to both sides, leaving me with X cannot equal positive seven. So now you can see that we have these two ranges. So I'm gonna go ahead and draw a number line. I'll say this is -9. Call this zero. I'll say this is seven. So my first range is saying that X can equal all values Except for -9 sold drawl a circle at -9. My second range is saying that X can equal all values except positive seven. So I'll draw another circle at seven. So now you can see that this number line sort of represents all the values of X. That will result in a real number. So let's translate this number line into interval notation. So from here you can see that X can come from negative infinity All the way to -9. And then from negative nine it can go all the way to seven And then again it can go from 7 to positive infinity. So notice how I use parentheses for my numbers. So wherever I have negative nine or seven I'm using parentheses because I know that X cannot equal negative nine or seven. So this becomes my final domain for my equation. And this ensures that any value of X that falls within this range and it gets plugged into my F. Times G equation will result in a real number and this aligns with answer choice C. So hopefully this video helped and see you next time

In Exercises 31–50, find fg and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

Key Concepts

Function Composition

Domain of a Function

Rational Functions

Watch next