BAM!!! So you want to identify what domain and range is when you look at the graph of
an equation? When then you are going to have to know the definitions. The set of all possible
input values, usually the x variable, which produce a valid output from a given equation.
Or you can say it is the set of all first elements in ordered pairs, well ordered pairs
are a grouping of two numbers the domain and range. The left and right movement and the
up and down movement. Of if you are talking about your graph being on the Cartesian Plane,
that would be the x the movement in terms of the x axis and the y value or the movement
along the y axis. The x and y values included in the graph on the Cartesian Plane. Usually
it is x comma y unless you are looking at some kind of graph that is displaying a real
life scenario where there is some real context to what the horizontal and vertical axis are.
Such as the age being along the horizontal axis or the x axis and the your height being
along the y axis. I am saying this again because I am going to make this lesson you are watching
right now an excerpt of a full length lesson. The range is the set of all output values
from an equation. It is very difficult to determine what the range is from an equation
unless you are very knowledgable, you have strong knowledge of the parent functions look
like. Like if you know what a standard line looks like, a standard parabola, a standard
linear absolute value function, maybe even a square root function and such. If you understand
and know what those parent functions look like and you are very knowledgable of translations/transformations,
and this lesson is for an Algebra class and we have not even got to that yet. I would
strongly suggest you make a graph from that equation first before you try to determine
what the range is. The set of all output values from an equation. The set of all second elements,
usually the y values, in ordered pairs. If you are talking about domain and range type
organization, that ordered set, that is usually an x and a y value. So let's move on to our
three examples. We have here an ellipse and the domain is again remembering that this
graph is made up of an infinite number of points each of those individual points having
there....well I guess there is a couple of points with the same x value but really each
of those points having their own value of x...their own coordinate right... That point,
that coordinate describes how move from the origin to that point. There is some kind of
left right movement, a negative x or positive x, and some up and down movement a negative
y or a positive y. So, what values of x are included in all of these individual points?
Well domain, looking at that horizontal movement, is going to be the smallest value of x that
is included looks like goes as far to the left as negative three and the as far to the
right as positive three. This is a smooth, actually being smooth is nothing important.
This is a continuous graph so every single x is included like negative three, negative
2.99999999, negative 2.9, -2.8, -2.7, -2.6999953821.... Any x that you can say that is between negative
three and positive three including those values will be part of the domain. The range then would be the output values if I were
looking at an equation, but graphically again this is the x axis and the y axis. The range
are all the y values that are included in that graph. Now again this graph is continuous
so it seems like it goes as far down as negative two and as far up as positive two. Every single
y value is included between that negative two and positive two. The range is going to
be greater than or equal to negative two and less than or equal to positive two. I am writing
these in inequality notation. In a PreCalculus book you might see this in interval notation.
I am not liking the fact that color does not match now. In number two... My air-conditioner
just came on and I know that bothers the video sometimes so let me shut that off and come
right back. Now that is better. Our second graph is a collection of five points. If I
want to list the domain here then I really just need to write a list of values for x.
Because this is a discrete graph. CHALK DOWN!!! I can see that there is only five points.
We have an x value of negative one, an x value of zero, one, two, and three. My y values
are... Now I don't need to repeat the same y value over and over again. I can just see
that this point has a y value of negative two, and it is good practice to write these
from low to high, and then we have a y value... from the origin we need to go down one so
my y value is negative one. Finally we have a y value of positive one. That is how you
list domain and range for a discrete graph. Just list all the individual x and y values.
Now if I where to come back here and change this to a graph that goes on forever and is
continuous and not just focus on those five discrete points, now my domain is going to
be... I intended on writing these in different colors. Now my domain, well this graph is
going up but it is also spreading to the left and right forever, so the domain is all real
numbers. You can say that x is all reals. You might have a textbook that uses a double
R for all reals. You may say that x is greater than negative infinity and less than positive
infinity because again it goes on forever. You cannot say equal to infinity because that
is an infinite concept, it goes on for ever and ever and ever! And the range, well the
smallest... Again all these points are in here now, there is an infinite number of points.
The smallest y value is, or the point that is the farthest down on this graph and from
here the graph goes up forever, the smallest y value is negative two. The range is going
to be y is greater than or equal to negative two and less than infinity because that parabola
is opening up forever. The y values are just going to keep getting bigger and bigger and
bigger. But the smallest y value included is a concrete value of negative two. Now for
our bottom graph that seems to go left and right forever. There are no breaks in this
graph except at two. So our domain is x cannot be equal to two. You may write that x is greater than negative infinity and
less than two and x is greater than two and less than positive infinity if your teacher
wants you to write it that way. Now the range values.... Get down with it:D The range values
are all the y values that are included in the graph. The graph is no where...no where
do you see a purple line below this horizontal asymptote, this horizontal line that is at
y equal is one. So these y values are going to get small and smaller, maybe here it is
two, maybe it is at 1.5, 1.1, 1.0001. Now I am on the left side where the x's are negative
but my y is still positive. And the y's are going to get closer to closer to that y value
of one but never really become one. So we are going to say the range is y is greater than one and less, now this goes up
forever, so y is never going to stop growing so y is less than infinity. In an upcoming
algebra lesson we are going to be talking about finding min and max's and the parabola
has a minimum y value of negative two, but because it goes up forever and never stops
going up, we cannot say there is a maximum value unless you want to say the maximum value
is infinity. But when we talk about finding maximum and minimums in a graph, we are looking
for a concrete like a Real answer if you will. So, this graph has a min of negative two and
max of positive two. This function, this parabola, has a minimum of negative two but has no maximum.
And this graph is approaching one but it is becoming infinitely and infinitely closer
to one and coming down from above it, but because there is no concrete number that the
graph becomes this graph does not have a minimum or because it goes up forever it does not
have a maximum. Again, if you want to talk about finding domain of an equation as opposed
to finding domain from a graph, I will have a link to my video lesson above my head or
down in the description if you not using Flash because you are watching on an iPad or something
like that. I am Mr. Tarrou. BAM!!! Go Do Your Homework.