Functions – Domain
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Functions – Domain

(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.

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