In the following graph, what is 𝑓 of negative four?
So to read off 𝑓 of negative four from the graph, we need to find negative four on the 𝑥-axis — there it is — and then we need to move vertically from this point on the 𝑥-axis until we hit the graph. If we go up a bit, we hit this hollow circle with 𝑦-coordinate one.
So is the value of 𝑓 of negative four that we’re looking for one? Well, it’s not clear because if you go up further, we hit another hollow circle with 𝑦-coordinate three. And going up further, we hit this filled circle with 𝑦-coordinate four.
So which of these three circles gives us the value of 𝑓 of negative four? It turns out that it’s the filled-in circle with coordinates negative four, four. 𝑓 of negative four is then the 𝑦-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 𝑓 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 𝑓 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 𝑓 of negative four is not three.
We’ve seen that the point negative four, four is on the graph, and that’s what the filled-in circle tells us. And as usual, for a graph, the 𝑦-coordinates of this point gives us the value of the function.