We want to determine the domain and range of a function given the graph of the function. The domain is a set of all possible x values of the function. X values occur along the x axis or the horizontal axis.
And the range is a set of all possible y values of the function and y values occur along the vertical axis. So if we're given the graph of a function and we want to determine the domain of the function, We want to project the graph onto the x-axis or determine how the graph behaves horizontally along the x-axis. What I mean by that is notice how the leftmost point of this graph occurs right here when x is approaching negative three and the rightmost point on the graph would be here when x is equal to positive two and the graph would also contain every x value between negative three and two.
But there's one more thing we need to be careful about here. Negative three is not going to be in the domain of this function because of this open point here. So let's make an open point here to indicate that. But notice that x equals two, this point is closed, so it would include positive two.
So the domain of this function is going to be from negative three to positive two, not including negative three, but including positive two. So if we want to express this using inequalities, we would say x is greater than negative three and less than or equal to positive two. If we want to use interval notation, the interval is from negative three to two. It includes two, so it's closed on two, so we use a square bracket.
And it's open on negative three because it does not include negative three, so we use a rounded parenthesis. These two mean the same thing. And then to determine the range, we now want to project this function onto the y-axis, or determine how it behaves vertically. So again, notice how the lowest point on this graph here is approaching negative five, and then it includes every y value all the way up to this high point when y is positive five.
But notice how it's not going to include negative five because of this open point. but it will include positive five because of this closed point. So the range is going to be from negative five to positive five, not including negative five, and including five.
So we can say y is greater than negative five and less than or equal to positive five. We're using interval notation, square bracket for five because it includes five, and a rounded parenthesis for negative five because it does not include. negative five.
Now let's go ahead and take a look at a second example. We'll start by determining the domain. So we're going to project this function onto the x-axis or determine how it behaves horizontally.
The leftmost point occurs right here at x equals negative four and then notice how the graph moves to the right indefinitely because we are assuming this graph is going to continue in this direction. So the domain would start at negative four and then move to the right indefinitely, meaning it's going to approach positive infinity. So the domain would be x is greater than or equal to negative four. We're using interval notation.
We have the interval from negative four to infinity. It includes negative four, so we have a bracket. And then for infinity, we always use a rounded parenthesis. And then for the range, we want to project this function onto the y-axis, or determine how it behaves vertically.
So the lowest point on this graph is right here at y equals negative 4. This is a closed point, so it does include negative 4. Now when we try to determine how high this graph goes, we need to be careful because, of course, it is moving to the right very fast, but notice how it also is moving upward. So even though it's not shown on the screen, this graph would continue to move upward. and therefore the range is going to approach positive infinity.
So the range would be y is greater than or equal to negative four. We're using interval notation. Just like for the domain it'd be closed on negative four to positive infinity.
Okay so I hope these two examples were helpful.