Video Transcript
In the following graph, what is ๐ of negative four?
So to read off ๐ of negative four from the graph, we need to find negative four on the ๐ฅ-axis โ there it is โ and then we need to move vertically from this point on the ๐ฅ-axis until we hit the graph. If we go up a bit, we hit this hollow circle with ๐ฆ-coordinate one.
So is the value of ๐ of negative four that weโre looking for one? Well, itโs not clear because if you go up further, we hit another hollow circle with ๐ฆ-coordinate three. And going up further, we hit this filled circle with ๐ฆ-coordinate four.
So which of these three circles gives us the value of ๐ of negative four? It turns out that itโs the filled-in circle with coordinates negative four, four. ๐ of negative four is then the ๐ฆ-coordinate of this point four.
When trying to find the value of a function from its graph, if you have a choice between a filled circle and one or two hollow circles, then the value always comes from the filled circle.
So why are the hollow circles there? What do they tell us? This hollow circle with coordinates negative four, one should be seen as the end of the ray pointing to the left. You can think of it as circling the empty space at the point negative four, one, emphasizing that this point is not on the ray and so is not on the graph. This ray tells us that the value of ๐ of some number less than negative four is one.
In a similar way, the other ray which points to the right tells us that the value of ๐ of some number greater than negative four is three. Again, the hollow circle at the coordinates negative four, three emphasizes that this point is not on the ray and so not on the graph. In other words, that ๐ of negative four is not three.
Weโve seen that the point negative four, four is on the graph, and thatโs what the filled-in circle tells us. And as usual, for a graph, the ๐ฆ-coordinates of this point gives us the value of the function.
