### Video Transcript

Simplify the function ๐ of ๐ฅ is equal to seven ๐ฅ plus 63 over ๐ฅ plus six over ๐ฅ
plus nine over ๐ฅ plus six and find its domain in the set of real numbers.

Okay. So first of all, weโre gonna have to do a bit of algebra to simplify this
function here, and then we can go on to consider the domain, the set of valid input values to
that function.

So first, letโs try to simplify this denominator here. So weโre adding two terms, ๐ฅ and nine over ๐ฅ plus six. Now the first term is ๐ฅ and the second term is a fraction. So I can to- convert
this into a fraction by just putting ๐ฅ over one. So Iโve now got fraction plus another
fraction. And by multiplying that first term by ๐ฅ plus six over ๐ฅ plus six, which is just
equal to one so itโs not changing the size of the number, Iโm gonna create a fraction which
has got the same denominator as the other fraction. So just swapping around the ๐ฅ and the ๐ฅ plus six on the numerator there, that
first term becomes ๐ฅ times ๐ฅ plus six over ๐ฅ plus six. And the second term is still nine over
๐ฅ plus six. Now Iโve got a common denominator. I can just add the numerators together. And that gives me ๐ฅ times ๐ฅ plus six plus nine over ๐ฅ plus six. Now I can multiply out the parentheses, ๐ฅ times ๐ฅ gives me ๐ฅ squared and ๐ฅ times
positive six gives me positive six ๐ฅ. And that gives me ๐ฅ squared plus six ๐ฅ plus nine over ๐ฅ plus six.

Now we can also notice that ๐ฅ squared plus six ๐ฅ plus nine factors down to ๐ฅ
plus three times ๐ฅ plus three. And itโs always a good idea to factor as much as we can because then hopefully
some things might cancel. So now weโve got this new expression here, which is equivalent to that
expression there. So letโs just copy that across. So Iโve converted my denominator here into an expression, where Iโve got two
terms multiplied together divided by another term. And hopefully, if we do a similar process to the original numerator here, weโll
generate an expression which has got things which will cancel out with our denominator down
there. So letโs go ahead and have a go at that.

So weโll start up with seven ๐ฅ plus 63 over ๐ฅ plus six. I can convert the first
term into a fraction. So Iโve now got a fraction plus another fraction, and I need to create a
common denominator. So I can multiply the first term by ๐ฅ plus six over ๐ฅ plus six, which again is
one. So one times seven ๐ฅ over one is still just seven ๐ฅ over one. And that gives me seven ๐ฅ times ๐ฅ plus six over ๐ฅ plus six plus 63 over ๐ฅ plus
six. Now Iโve got a common denominator. I can just add the numerators together again. And it gives us seven ๐ฅ times ๐ฅ plus six plus 63 all over ๐ฅ plus six. Now I could multiply seven ๐ฅ by ๐ฅ to give me seven ๐ฅ squared and seven ๐ฅ by
positive six to give me plus 42๐ฅ and then have plus 63, but Iโm actually not gonna do that in
this occasion. Because Iโve spotted that seven is a factor of 63, so I can factor out a seven. So weโve got seven here and weโve got seven times nine is 63. And when we work that out, weโve got seven times ๐ฅ times ๐ฅ plus six plus nine
over ๐ฅ plus six.

Now the reason I did all that and factored out the seven, is because the
contents of these parentheses here, ๐ฅ times ๐ฅ plus six plus nine, thatโs exactly the
expression that we had before in the- when we were simplifying the denominator. So when I multiply ๐ฅ times ๐ฅ and ๐ฅ times positive six, that gives me ๐ฅ squared plus six ๐ฅ. So weโve got ๐ฅ squared plus six ๐ฅ plus nine
in the parentheses there over ๐ฅ plus six, and we saw that ๐ฅ squared plus six ๐ฅ plus nine
factored down to ๐ฅ plus three times ๐ฅ plus three. So weโve got seven times ๐ฅ plus three times ๐ฅ plus three over ๐ฅ plus six.

So when we simplified that numerator, weโve come up with an expression here
which is actually very similar to the denominator that we came up with. So weโve managed to rewrite our original expression for the function ๐ of ๐ฅ as
this expression over here, which doesnโt look particularly as though weโve simplified it.
But remember, weโve got this numerator divided by this denominator. And when we write that out, weโve got the function ๐ of ๐ฅ is equal to seven
times ๐ฅ plus three times ๐ฅ plus three over ๐ฅ plus six divided by ๐ฅ plus three times ๐ฅ plus
three over ๐ฅ plus six.

And when we divide fractions, we just simply have to flip the second fraction and
turn it into a multiplication. So that becomes seven times ๐ฅ plus three times ๐ฅ plus three over ๐ฅ plus six
times ๐ฅ plus six over ๐ฅ plus three times ๐ฅ plus three. Now that turns out to be quite handy because weโve got lots of terms multiplied
together on the top and lots of terms multiplied together on the bottom. But the convenient
thing is that some of the terms on the top are the same as some of the terms on the bottom. So
weโre gonna be able to cancel them out.

For example, dividing the top by ๐ฅ plus six gives us one here. Dividing the bottom
by ๐ฅ plus six gives us one here. Dividing the top by ๐ฅ plus three here gives us one. Dividing
the bottom by ๐ฅ plus three here gives us one. Dividing the top by ๐ฅ plus three gives us one.
Dividing the bottom by ๐ฅ plus three gives us one. So weโve got seven times one times one times one
over one times one times one.

Hey presto! This great brick expression that we had right at the very beginning, for
n of ๐ฅ, simplifies down just to ๐ of ๐ฅ is equal to seven. So thatโs the answer to the first part of the question. The simplified version of
that function is ๐ of ๐ฅ is equal to seven.

Okay. Now letโs consider the domain of this function. What are the valid ๐ฅ-values to
go into that function? Well, on the face of it, you might think that itโs just any real number because
the function is always just equal to seven. But wait, thatโs not quite right. If we look at our original expression for the
function over here, if this denominator here turns out to have a value of zero, then weโve got
something divided by zero and that will be undefined. So an ๐ฅ-value that generates a denominator in
that function of zero would need to be excluded from the domain. And likewise, up here on the
numerator, weโve actually got this term 63 over ๐ฅ plus six. If the value of ๐ฅ that we
put into the function generates a value of zero in this position here, weโre gonna have
63 divided by zero which again is gonna be undefined.

So we need to ask ourselves, which ๐ฅ-values must we exclude from the domain? Well, the value of ๐ฅ that makes this bit equal to zero โ in other words, when ๐ฅ plus
six is equal to zero โ needs to be excluded. And subtracting six from both sides of that equation tells us that ๐ฅ would need
to be negative six for that to happen. So thatโs the first value of ๐ฅ that we need to exclude from the domain.

But we also said that if the denominator here turns out to be zero, then whatever
value of ๐ฅ that makes that equal zero also needs to be excluded. So ๐ฅ plus nine over ๐ฅ plus six
is equal to zero. And remember that we already reexpressed that in this way, ๐ฅ plus three times ๐ฅ
plus three over ๐ฅ plus six. So whatever value of ๐ฅ makes that equal to zero is a value that we need to
exclude as well. Well, in order for this expression here to be equal to zero, we need to make the
numerator equal to zero. And both terms multiplied together there are in fact ๐ฅ plus three.

Now it doesnโt matter whether we have ๐ฅ plus three being equal to zero, or ๐ฅ
plus three being equal to zero. So long as one of those sets of parentheses is zero, then those
two things multiplied together are gonna give an answer of zero. Since theyโre both the same, ๐ฅ plus three, itโs basically ๐ฅ plus three that has to
be equal to zero. And subtracting three from each side of that equation tells us that ๐ฅ would need
to be equal to negative three in order to make that expression equal to zero. So we can exclude ๐ฅ is equal to negative three from our domain as well.

So the domain is equal to the real numbers but excluding negative six and
excluding negative three as well. And one way that we can write that out is to say that the domain is equal to the
set of real numbers minus the set containing negative six and negative three. So there are our
answers.