Video: Simplifying and Determining the Domain of a Difference of two Rational Functions

Simplify the function 𝑛(𝑥) = 3𝑥/(𝑥 + 4) − 7𝑥/(𝑥 − 4) and determine its domain.

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Video Transcript

Simplify the function 𝑛 𝑥 equals three 𝑥 over 𝑥 plus four minus seven 𝑥 over 𝑥 minus four, and determine its domain.

To solve this kind of problem, there are two parts. We have to simplify the functions and determine the domain. We’re gonna start with looking at the domain. So before we can determine the domain, we need to say, “Well, what is the domain?” Well I like this definition: the domain is the complete set of possible values of the independent variable. Okay. Nice. But still, what does that mean?

What it actually means is we want to know what values of 𝑥 make the function work and will give real values of 𝑦 or, in this case, 𝑛 𝑥. Examples of when a function might not work would be for instance if we had the square root of a negative number, so that would make the function not work, or if we had zero as our denominator. That would also cause the function to not work.

And it’s this second case we’re gonna look at today because we’re going to see which values are gonna make our denominators equal to zero. Okay, let’s try and solve this problem. So we want to determine the domain of our function. To do this, we set the denominators in our function to zero.

We’re gonna start with the first term, so the left-hand term. I’m gonna make the denominator of that equal to zero. This gives us 𝑥 plus four equal to zero. Now we can solve this by subtracting four from each side, which gives us a value of 𝑥 is equal to negative four. So great! This is the first value of 𝑥 that won’t allow our function to work.

Now we’re gonna have a look at the second term in our function. And so 𝑥 minus four, we’re gonna make that equal to zero. Again, we’ll just solve that. So we’re gonna solve It by adding four to each side, which gives us an answer of 𝑥 is equal to four. So great! We’ve now found the two values of 𝑥 which will not allow our function to work.

So now we can express our domain. And we do so like this. Because we say that our domain is equal to 𝑟, which means the real numbers so all the real numbers, minus negative four and four. So that means the domain is all real numbers except negative four and four.

Great! Now we can move on to the second part of this question. So now we’re gonna simplify our function. In order to simplify this function, we need to have a look at it. And we’ve actually got two terms that are both fractions. And actually what we’re trying to do here is subtract fractions. And we’re gonna do in the same way that we’d always do it, and that’s to find a common denominator.

To create that common denominator, we’re gonna multiply our first term by 𝑥 minus four, and that’s the numerator and denominator, which gives us three 𝑥 𝑥 minus four over 𝑥 plus four 𝑥 minus four. And then we’re gonna multiply our second term by 𝑥 plus four, so the numerator and denominator, which is gonna give us seven 𝑥 𝑥 plus four over 𝑥 minus four 𝑥 plus four.

Brilliant! So now we can see that we’ve actually got the same denominator for each term in our function. Now for the next stage, what we want to actually do is to rewrite it because we want to rewrite it as one single fraction. And we can do that because we’ve now got the common denominator in each of our terms.

And as you can see I’ve identified the fact that each of these denominator are the same. It doesn’t matter if they’re written the other way round. They’re the same denominator. Great! Now we want to move onto is actually simplifying our numerator. But to enable us to do that, we can’t actually do that the moment. So what we’re gonna do is we’re actually going to expand the parentheses. We’ll start with the first parenthesis on the numerator.

So we’ve got three 𝑥 multiplied by 𝑥, which would give us a three 𝑥 squared. Then three 𝑥 multiplied by negative four, which would give us negative 12𝑥. Then we can do the second parentheses. So we’ve got negative seven 𝑥 multiplied by 𝑥, which would give us negative seven 𝑥 squared. And then, finally, negative seven 𝑥 multiplied by positive four. But remembering to be very careful with our negatives here because what that would give us isn’t positive 28𝑥.

It actually gives us negative 28𝑥. This is one of most common mistakes. Be careful of this. Great! So we’ve expanded our parenthesis. Now let’s get on with the simplifying. Okay. Now to simplify the numerator a bit further, what can we actually do is we can actually look at collecting our like terms. So we’ll do that now. Firstly, there’s three 𝑥 squared minus seven 𝑥 squared, which would give us negative four 𝑥 squared.

Then we can do negative 12𝑥 minus 28𝑥, which gives us negative 40𝑥, remembering again to be careful with our negative numbers. I know it sounds simple. But honestly, it’s one of the most common mistakes at this stage. And then this is all over our 𝑥 plus four 𝑥 minus four.

At this stage you’re probably thinking to yourself, “Right. Okay. Are we close to simplifying? Is this is fully simplified?” So what I’d say now is have a look at our numerator. Okay, we can’t collect any more like terms. But can it be factored? Always check. Can it be factored? And if it can, let’s factor it.

In this case, we can look at the coefficients of 𝑥 squared and 𝑥. And we can see that they can both divide by four and they can both divide by negative four. So it’s going to be our first factor is negative four. Then we look at the 𝑥 squareds and the 𝑥. We actually see that actually there’s an 𝑥 in both the terms.

So our other factor outside the parentheses is going to be 𝑥. So now we’ll look at the terms that will go inside. So we’ve got negative four 𝑥. If we times that by 𝑥, that would give us negative four 𝑥 squared. And then if we’ve got negative four 𝑥, what do you need to multiply that by to get to negative 40𝑥? Well that’s gonna be positive 10.

Fantastic! So now actually looking at that, we can think, “Right. Is-is that fully simplified?” And the answer is well can we do anymore to our numerator. Can we factor any further? Can we collect anything? No. So yes it’s fully simplified. But for our final answer, the way I’d actually leave it is in this kind of way.

And to use this form, just pull it because the negative actually applies to the whole of our fraction. Great! So we’ve now finished the problem. And we’ve simplified the function. And we’ve determined its domain. Quick reminder: the domain is any of the complete set of possible values of 𝑥 that would actually give us a function that works.

So we found that in this case by making each of our denominators equal to zero and then solving. And then simplifying the function, again, if it’s fractions that we’re looking at, treat them just the same as any fraction even if they’ve got algebraic terms within them. And so in that case, find your common denominator and then subtract our numerators.

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