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(male narrator) In this video,
we're going to begin taking a look
at what is called the "domain of a function."
Domain is simply a list of all the possible...
input...
or x-values.
In other words, a domain is a list
of what numbers we are allowed to plug into a function
without making the function undefined.
There are two types of functions that we should be aware of
that can create an undefined situation.
One of the key ones is a fraction.
In fractions, it is important to remember
that what is not allowed to occur
is that the denominator...
can't...be...0.
If the denominator ends up being 0,
the function would be undefined.
Another situation we must be aware of is even radicals.
If the index is even, what is inside cannot be negative.
It must be greater than or equal to 0.
We can't take an even root of a negative number.
This would make the function undefined.
Let's try some examples where we find the domain
or list what values keep the function from being undefined.
In this problem, while there are several things occurring,
what is interesting to us is the 4th root.
The multiplying by 3 doesn't interest us as much--
as we can multiply any number by 3,
and we can add 4 to any numbers.
There's no restrictions.
But we do have a restriction on a 4th root,
and that is the stuff inside the radical.
The 2x minus 6 can't be negative.
It must be greater than or equal to 0.
Solving this equation by adding 6 to both sides,
giving us 2x is greater than or equal to 6,
and then dividing by 2,
gives us the domain of the function
to be x is greater than or equal to 3.
In other words,
we're saying that any value greater than or equal to 3,
when plugged in for x, will give us a real solution.
If we pick a number smaller than 3 and plug it in for x,
we'll end up with a negative value under the root,
which makes the function undefined.
Let's take a look at another example.
In this problem, we have several things happening,
but notice, as we look
for restrictions on what happens,
we can multiply any number by 2.
We can add 7 to any number.
We can take the absolute value of any number.
We can also square any number.
We can multiply anything by 3,
and we can subtract 4 from anything.
As we look through the operations in this function,
there are no restrictions that are not allowed to occur.
Because there are no restrictions
on the domain or the input value,
we say the domain here is all real numbers.
In other words, any number you plug in for x
will make this equation a true equation...
that has no undefined points.
Let's try one more example.
This problem's a fraction.
You may remember with fractions, the denominator cannot equal 0.
With fractions, the numerator can be anything it wants,
and so, when we're looking for the domain,
we don't really care about the numerator.
When we're looking for domain,
we're gonna focus on the denominator
and find out when it equals 0.
We can quickly find this by factoring
to x, minus 2, times x, plus 1, knowing that cannot equal 0.
Setting each factor equal to 0--
x minus 2, equals 0; and x plus 1, equals 0--
we can quickly solve by adding 2 to get x equals 2,
and subtracting 1 to get x equals -1.
We'll say that x cannot equal 2, and x cannot equal -1.
Because if we were to plug those values into this function,
we would end up with 0 in the denominator,
and that would be undefined.
Domain is a list of what keeps the function defined,
whether we're saying what x does not equal
or whether we're saying what x is greater than.