We want to determine the domain and range of a function
given the graph of the function. The domain is the set of all possible x-values
for the function, and the range is the set of all possible y-values for the function.
X-values are along the horizontal axis, and y-values are along the vertical axis.
So, to help us determine the domain of this function, we want to project this function
onto the x-axis or analyze it to determine how the graph behaves from left to right. If we were
to project this function onto the x-axis or if we had a bright light up here that was shining down
on the function, it would produce a shadow on the x-axis from negative two to positive four.
But notice how we have an open point here at x equals negative two, so it would not include
the value of negative two, so we'll make an open point there. But because this point is closed,
it would include positive four. So, the domain of this function would be the interval from
negative two to positive four, not including negative two but including positive four.
We can express this in a couple of ways. Using inequalities, we can say x is
greater than negative two and less than or equal to four. Or, if we want to express
this using interval notation, it would be the interval from negative two to positive four.
Since it's closed on positive four, we use a bracket, and since it's open on negative two,
we use a rounded parenthesis here.
Now for the range, we're going to do the same type of analysis, but now we want to project
the function onto the y-axis or analyze the graph to determine how it behaves vertically.
The lowest point on this graph would be right here at y equals zero, and then notice how it would
extend all the way up to y equals positive eight. In this case, it does include zero,
and it also includes positive eight, so the range would be the closed interval from zero to eight.
Again, we can express this using inequalities as y is greater than or equal to zero and
less than or equal to eight. Or, using interval notation, we have the interval from zero to eight,
and it's closed both on zero and eight because it would include the endpoints.
Now, let's take a look at another example. Again, we'll first try to determine the domain,
which is the set of all possible x-values, and x-values are along the horizontal axis. So, we
want to project this function onto the x-axis or analyze it to determine how it behaves moving from
left to right. The leftmost point on this graph would be right here at x equals negative two,
and then it would include every x-value all the way out to positive two. But notice how
we have an open point right here at x equals two, so it does not include positive two. So,
it's an open point here, but it does include this point here, so it's closed here. So, the domain
would be the interval from negative two to two, including negative two and not including two.
We could say x is greater than or equal to negative two and less than positive two,
or using interval notation, we have the interval from negative two to two,
closed on negative two and open on two.
Now, to determine the range, we want to project the function onto the y-axis
or the vertical axis or analyze the function to see how it behaves vertically.
The lowest point on this graph would be right here at y equals negative four. Notice how it
would include negative four because this point is closed, and then the graph includes every y-value
all the way up to positive four, not including positive four again because of this open point.
So, the range would be the interval from negative four to four,
including negative four but not including positive four. So,
we'd have y is greater than or equal to negative four and less than four using inequalities,
or using interval notation, from negative four to four, open on four and closed on negative four.
Okay, I hope you found these two examples helpful. We'll take a look at two more in the next video.