Combinations of functions and their domains (KristaKingMath)
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Combinations of functions and their domains (KristaKingMath)


Hi everyone. Today we’re going to talk about
how to calculate combinations of two functions and then define the domain of those combinations.
To complete this problem, we will calculate our combinations, simplify and then define
the domain of each. Let’s take a look. In this particular problem, we’ve been given
the two functions f of x equals x cubed plus 2 x squared and g of x equals 3 x squared
minus 1 and asked to find the following four functions and their domains. So as you can
see, we’re going to be finding f plus g, f minus g, f times g and f divided by g and
we’re just talking here about combinations of functions. These are different combinations of f and
g. The first one is obviously just the sum of the functions, the difference of the functions,
the multiplication and the division of the functions or the product and the quotient
of the functions. So let’s go ahead and start with f plus
g of x and this is going to be really simple. When we add functions together like this to
find a combination of these functions, we will just add f and g together. So f here
is x cubed plus 2 x squared and g of x is 3 x squared minus 1. So we will add 3 x squared
minus 1 and now we just need to simplify that. As you can see, we will get x cubed. When
we add 2 x squared and 3 x squared together, we will get 5 x squared and then of course
our negative 1. And that’s it. That’s really as far as we can simplify. We’ve
also been asked to give the domain of each of these combinations of functions. So we
will go ahead and do that and we will say that the domain here, the domain is equal
to – the domain of any polynomial function like this is all real numbers so we can just
denote that with all real numbers like that. As far as the difference of the two functions,
really similar, we have f minus g of x here. So we will just take f, just x cubed plus
2 x squared and we will subtract g of x. Don’t forget to put g of x in parenthesis like that
so that you make sure you distribute the negative sign. So when we subtract here, we will get x cubed.
2 x squared minus 3 x squared will give us a negative x squared and then we have minus
negative 1 which will give us a plus 1. Again, we’ve really simplified as much as we can.
So now we just need to indicate the domain and in this case, again we have a polynomial
function so the domain is all real numbers. If we look here at the product of these two
functions f times g of x, similar concept, we will just be multiplying the two together
so x cubed plus 2 x squared and then multiply that by g of x which is 3 x squared minus
1. So now we will be expanding these with our FOIL method. So we will have x cubed times
3 x squared which will give us 3 x to the fifth. x cubed times a negative 1 will give
us negative x cubed. 2 x squared times 3 x squared will give us plus 6 x to the fourth
and 2 x squared times the negative 1 will give us a negative 2 x squared. So now when we reorder our terms, we will
get 3 x to the fifth plus 6 x to the fourth minus x cubed minus 2 x squared and again
we have a polynomial which means that our domain is all real numbers. Finally our last function here, the quotient
of f and g, we will take f of x and we will divide it by g of x. So we will see that our
numerator here will be f of x which is x cubed plus 2 x squared and we will divide that by
g of x which we know to be 3 x squared minus 1 and there’s really not a whole lot we
can do here to simplify. We could factor out an x squared from our numerator but that wouldn’t
allow us to cancel anything. So this will be our quotient function. We just need to
figure out what the domain is. Remember that the domain of a rational function
like this, rational function being the quotient of two polynomials, a polynomial and a numerator
and a polynomial and a denominator, is undefined wherever the denominator is equal to zero.
So the domain is going to be all real numbers except where the denominator equals zero. So we need to figure out where that is the
case. So we will say over here, 3 x squared minus 1 equal to zero, and we need to solve
this to figure out which part of our domain is undefined. So we will add 1 to both sides to get 3 x
squared equals 1. Divide both sides by 3 to get x squared equals one-third and then we
want to take the square root of both sides to solve for x. So we will get x equals the
square root of one-third. So we will say the square root of one-third
and remember that when we’re taking the square root of a fraction like this, of a
quotient, we can take the square root of the numerator and the denominator separately.
So we will get x equals the square root of 1 over the square root of 3. We know that
the square root of 1 is just 1 and then we have the square root of 3 here in the denominator. Remember we have to include a positive or
negative because we could get positive 1 over the square root of 3 or negative 1 over the
square root of 3. Both of those would make our denominator equal to zero and therefore
our quotient undefined. So we can say that the domain of this function
is equal to all real numbers except x equals positive or negative 1 over the square root
of 3. So we can just say the domain is x not equal to positive or negative 1 over root
3 and that indicates that x can be any value except these two and defines our domain. That’s
it. That’s how you find the combinations of functions and their domains. So I hope you found that video helpful. If
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