Example visually evaluating discrete functions

What we have here is a visual depiction of a function, and this is a depiction of y is equal to h of x. Now, when a lot of people see function notation like this, they can see it as somewhat intimidating until you realize what it's saying. All a function is just something that takes an input. In this case, it's taking x as an input, and then the function does something to it, and then it spits out some other value which is going to be equal to y.

So, for example, what is h of 4 based on this graph that you see right over here? Pause this video and think through that. Well, all h of 4 means is when I input 4 into my function h, what y am I spitting out? Or another way to think about it: when x is equal to 4, what is y equal to? Well, when x is equal to 4, my function spits out that y is equal to 3. We know that from this point right over here. So, y is equal to 3, so h of 4 is equal to 3.

Let's do another example. What is h of 0? Pause the video, try to work that through. Well, all this is saying is if I input x equals 0 into the function, what is going to be the corresponding y? Well, when x equals 0, we see that y is equal to 4. So it's as simple as that: given the input, what is going to be the output? And that's what these points represent. Each of these points represents a different output for a given input.

Now, it's always good to keep in mind one of the things that makes it a function is that for a given x that you input, you only get one y. For example, if we had two dots here, then all of a sudden we have a problem at figuring out what h of six would be equal to, because it could be equal to 1 or it could be equal to 3. So, if we had this extra dot here, then this would no longer be a function. In order for it to be a function, for any given x, it has to output a unique value. It can't output two possible values.

Now, the other way is possible. It is possible to have two different x's that output the same value. For example, if this was circled in, what would h of negative 4 be? Well, h of negative 4, when x is equal to negative 4, when you put that into our function, it looks like the function would output 2. So, h of negative 4 would be equal to 2.

But h of 2 is also equal to—we see very clearly there—when we input a 2 into the function, the corresponding y value is 2 as well. So it's okay for two different x values to map to the same y value; that works. But if you had some type of an arrangement, some type of a relationship where, for a given x value, you had two different y values, then that would no longer be a function. But the example they gave us is a function, assuming I don't modify it.