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