In this video
we will be naming and determining the domain
of various functions.
The domain of a function is a set of all possible inputs.
If we have a function of X
we can say the domain
is a set of all possible X values.
To determine the domain, there are two rules to consider.
Number one we cannot divide by zero.
And number two, in dealing with the set of real numbers
we cannot take the square root of a negative number.
The first function is F of X equals X squared
plus three X minus 10.
This is a polynomial function of degree two
which we call a quadratic function.
And now the domain is going to be the set
of all possible inputs or X values
and we can take any real number, square it
and add three times the real number
and then subtract 10
and therefore the domain is all real numbers
which using a notation is the open interval
from negative infinity to positive infinity.
Next we have F of X equals five X plus seven
which is a polynomial function of degree one
and therefore we call this a linear function.
And again for the domain
we can multiply five by any real number
and then add seven
and therefore once again the domain
would be all real numbers
which is the open interval from negative infinity
to positive infinity.
But take a look at these two graphs
because the domain is a set of all possible inputs
and X values,
we find the domain along the horizontal or x-axis.
And because this graph moves to the left and right
without any holes or breaks,
this does verify the domain is all real numbers.
So this would be the graph
of the domain of the given function.
And the same thing for the linear function,
it moves to the left and right
without any holes or breaks.
And therefore the domain is all real numbers.
Next we have F of X equals the square root
of the quantity 36 minus three X.
This is a radical function
or more specifically a square root function.
To determine the domain,
the radicand of 36 minus three X must be non-negative
which means to find the domain
we must set 36 minus three X
greater than or equal to zero.
If 36 minus three X is greater than or equal to zero
it will be non negative and the square root will be real.
And we solve the inequality
we subtract 36 on both sides
to isolate the variable term
which gives us negative three X,
greater than or equal to negative 36.
Now to solve for X, we divide both sides by negative three
remember when solving an inequality,
we multiply or divide both sides by a negative,
we must reverse the inequality.
So in simplifying here
we have X on the left,
we reverse the inequality
so we have X is less than or equal to.
On the right, we have negative 36 divided by negative three
which is positive 12.
This is a domain.
So the domain is X less than or equal to 12,
but we need this in interval notation.
So if we were to graph this,
interval includes 12
so we have a close point on 12
and then values get less than 12 moving to the left
approaching negative infinity,
and therefore using interval notation
the domain is from negative infinity to positive 12
because it includes positive 12,
we use a square bracket to the right.
So this is the domain using interval notation.
Next we have F of X equals a square root
of the quantity two X minus eight.
Again this is a radical function
or more specifically a square root function.
The radicand must be non-negative
and therefore to find the domain
we must set two X minus eight,
greater than or equal to zero,
we add eight to both sides giving us two X,
greater than or equal to eight
divide both sides by two.
Simplifying we have X is greater than or equal to
positive four.
And if we graph this,
we have a close point on four arrow to the right
where values get greater than four
approaching positive infinity
and therefore using interval notation,
we have the interval from four to infinity
with intervals close or includes four
so we have a square bracket to the left
and parenthesis to the right.
Looking at these two graphs,
we can see the domain of this first square root function
is less than or equal to 12
or from negative infinity to 12
and the domain of the second square root function
is X greater than or equal to four
or from four to infinity.
Next we have F of X equals the cube root
of the quantity four X minus six.
Again this is a radical function,
or more specifically a cube root function.
Remember we can take the cube root
of both a positive and negative number as well as zero
and therefore the domain of the cube root function
is going to be all reals
or the open interval from negative infinity
to positive infinity.
Next we have F of X equals 19
because the output or function value is always 19,
this is called a constant function.
And again, the input or X can be any real number
it's just the output will always be 19
because the input can be any real number
again the domain is the open interval
from negative infinity to positive infinity.
Looking at these two graphs
we can see the graphs move to left and right
without any holes or breaks
which graphically verifies the domain is all reals
for the cube root function as well as the constant function.
Next, we have F of X equals X cubed minus four X.
This is a polynomial function of degree three
which is called a cubic function.
The domain, we can take any real number, cube it
and subtract four times the real number
and therefore the domain is all reals.
Next we have F of X equals the quantity X plus one
divided by the quantity X minus six.
This is a rational function.
A rational function as a ratio
or quotient of two polynomial functions.
Or in our case, we have a ratio or quotient
of two linear functions.
And now to find the domain,
because division by zero is undefined,
and to determine what value of X
will make the denominator equal to zero
once we find that value,
we must exclude the value from the domain.
So to find the domain of this rational function
we first set of X minus six equal to zero.
Solving for X, we add six to both sides
giving us X equals six.
Which means the domain of the rational function
is going to be all real numbers except six.
So let's graph this on the number line.
You want to graph all real numbers except six.
So let's say this is zero and this is six.
To exclude six, we have an open point on six
and then we graph all the other real numbers to the right
and to the left.
So the right we approach infinity.
To the left we approach negative infinity
which means using interval notation
the domain is going to be from negative infinity to six.
It does not include six so we have a parenthesis
to the right of six.
The union, we use this U for union
and then from six to infinity.
Again not including six.
This is the domain using interval notation.
Let's look at these two graphs.
Again we can see, looking at the cubic function
the graph moves to the left and right
without any holes or breaks
which verifies the domain is all real numbers.
And now for the rational function,
notice how there is a break in the graph at X equals six
but then from there,
the graph moves to the right and left
verifying the domain is all reals except six.
And for our last two examples,
we have F of X equals X,
this is a polynomial function of degree one
and therefore we can call this a linear function.
But because the input and output are both X
more specifically this is called the identity function.
In here because the input can be any real number
which will be the same as the output,
the domain is all reals
which again using interval notation
is the open interval from negative infinity
to positive infinity.
And then finally we have F of X equals
the absolute value of X minus three
because of the absolute value,
this is called an absolute value function.
And because we can take the absolute value
of any real number and then subtract three,
once again the domain is all reals.
Let's take a look at these two graphs.
And you can notice both graphs move to the left and right
without any holes or breaks
which verifies the domain is all real numbers
for both functions.
I hope you found this helpful.