In Exercises 31–50, find f−g and determine the domain for each fu...

Question

In Exercises 31–50, find f−g 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 minus G and write the domain of that composite function and we're given that F of X is equal to nine X plus two divided by X squared minus 49. And G of X is equal to eight X plus nine divided by x squared minus 49. So in this case our first step is going to be to find f minus G. And in this case f minus G is the same as the F function F of x minus the G function G of X. And were given the values of F F X and G of X. So I'm going to go ahead and plug in. So F of X is equal to nine X plus two divided by X squared minus 49. And that's being subtracted by G of X wishes eight x plus nine divided by X squared minus 49. So in this case you can see that we're subtracting two fractions to get our new function. So we want to make sure that the denominators are the same and in this case they are. So we can go ahead and do our subtraction. So that leaves us with nine x plus two -8 x -9 divided by the common denominator, X squared minus 49. And we can go ahead and simplify the numerator, we have nine x minus eight X. So that leaves us with just one X. And then we have positive 2 -9. So that leaves us with negative seven and we're going to divide that still by the denominator X squared minus 49. And in this case we can simplify the denominator further by factoring it. So the denominator is a difference of squares. So it can be factored to be x minus seven and X plus seven. And we can go ahead and check that to make sure that we factored it completely. So X times X is x squared X times positive seven is seven, X negative seven times X is negative seven X. And negative seven times positive seven is negative 49. So you can see that these middle terms cancel out and we're left with just X squared minus 49. So we factored the denominator correctly and once we factor the denominator you can see that The numerator and denominator partially cancel out because we have X -7 in both of them. So X -7 divided by X -7 is one And our fraction becomes one Divided by X-plus seven. So when we do f minus G We actually just get one divided by X-plus seven. So now that we have our new composite function, we can find the domain. And in this case because we're working with a fraction the domain will be the range of X values that give us real numbers for our answer. So in fractions, the only time where we wouldn't get a real number is if the denominator is equal to zero because if the denominator is equal to zero, our answer is undefined. So we just want to make sure that the denominator does not equal zero. So in this case our denominator is X plus seven and it cannot equal zero. So if I subtract 7 to solve for this expression I'm left with X cannot equal -7. So this says that X can be any value except for negative seven. So if I write that in interval notation, I have the X can go from negative infinity to negative seven And then from -7 to positive infinity. And you might be thinking that this is the domain for our composite function F minus G. However, because it's a composite function, we have to carry over the domains from the F and the G function. So if we go back, you can see that both the F function and the G function have the same denominator, X squared minus 49. And again, we want to make sure that the denominator never equal zero because that would give us undefined answers. So if we solve for the domain of the F. And G function, I have X squared minus 49 cannot equal zero. So at 49 to both sides. So that leaves me with X squared cannot equal positive 49. If I take the square root of both sides, I have X cannot equal positive or negative seven. So you can see that the previous functions Have some similarity to the new function because the old functions also exclude -7. But they have an additional number that they exclude, which is the positive seven. So we want to make sure to include that in our domain. So with this new values that we need to exclude, We still have from negative infinity to negative seven And X cannot equal -7. So there's a parentheses but now X can equal positive seven either. So I'll go from negative seven to positive seven and I'll do one last union from southern to positive infinity. So now this interval range excludes both negative seven and positive seven. So this becomes my final domain range and f minus G is equal to one divided by X plus seven. So if I go back to my answer choices, you can see that see has the correct answers. So hopefully this video helped and see you next time.

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

Key Concepts

Function Operations

Domain of a Function

Rational Functions

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