Hey, everyone. So up to this point, we've been spending a lot of time focusing on functions. We've been looking at the graphs of functions and finding the domain and range. And in this video, we're going to be taking a look at some of the common functions. Now, these are the functions that are going to frequently show up throughout this course. So it's definitely important to make sure we're familiar with these functions and their graphs and just in general how they work. So let's take a look at the constant function. The constant function occurs when our function f(x)=c and c can be any constant number. Let's say, for example, our function f(x)=2. This would be a constant function because 2 is just a constant, and so notice on our graph that y=2 is constantly the value in all directions. Notice how this value expands to all the negative and all the positive x values. We can input any x that we want into this function. So we would say that the domain goes from negative infinity to positive infinity because we can have any x value we want to, but notice how the range is only going to be where our y value is 2. So y=2 is the only possible output we can get for this function. So that's the domain and range of the constant function, and keep in mind that this is just going to be whatever c is. So it's not always going to be 2; it could be 4, it could be negative 5 thirds. It's just going to be whatever c is.
Now let's take a look at the identity function next. The identity function says that f(x)=x, and this function tells you that whatever you put into the function, you're going to get out of it. Let's say that you put negative one in for x. Well, in this case, you're going to get negative one as your output. Now, let's say you put 50 in for x. Well, in this case, you're going to get 50 as your output. So, whatever you put in, you're going to get out. And because of this, we can say that the domain is all real numbers because we can put any number we want to infer it. We can put any of these negative x's or any of these positive x's into the function. But notice how we can also get any negative y's or any positive y's out of this function. So we would see that the range is also all real numbers. So that's the identity function.
Now let's take a look at the square function. And the square function forms this interesting shape called a parabola. The parabola is this bowl-like shape that you see on the screen here. Now the square function happens when f(x)=x2. And the domain for this function is going to be all real numbers. Now the reason for all real numbers is that notice how all of the negative x's and all of the positive x's are defined by this curve. This curve is continuously expanding up and to the left and right, so we include all of the x values. But what about our range? Do we also have all real numbers for the range? Well, it turns out we actually don't because notice even though all of the positive y's are included, we don't include any of these negative y's. The curve does not expand down below the graph. So we can say that our range will go from 0 all the way to positive infinity, and we do include the 0 because we could have a value right here where the origin is. But we can never be below 0, so we can never get a negative output for the square function.
Now let's take a look at the cube function. The cube function says that our function f(x)=x3, and notice for this graph that we have all of the negative x's and all of the positive x's included. So we would say that the domain is all real numbers. You can get any number, you can plug any number you want in for x, and notice how the range for the y values, we include all the negative y's and all the positive y's in this curve. So we can say that the range is also all real numbers. So that is the domain and range for the cube function.
Now let's take a look at the square root function. And the square root function has the most restrictions of all the common functions. So in this function, we have to look at our graph here. Notice how this graph continuously goes to the right and also goes up. And because it continuously goes to the right, we can see we have all positive x's, but notice how none of the negative x's are included. So we would say that the domain goes from 0, including 0, all the way up to positive infinity. Because we could have our square root function be 0, but we can't ever be below this; we cannot put a negative number in for x. Now let's take a look at our range. Notice that our range continuously goes up, so we have all the positive y values, but none of the negative y values are included. So we would say that our range also goes from 0 to positive infinity. So this is one of the functions where the x values cannot be negative because none of these negative x values are included by this curve.
Now let's take a look at the cube root function. The cube root function is where we have our function f(x)=x3 and notice for this function we have all of our negative x's and all of our positive x's included. So we can say that the domain goes from negative infinity to positive infinity. All real numbers are going to be included for the domain because we see that our curve goes to all real numbers. And if we look at our y values, we can see that all of the positive y's and all the negative y's will also be included because this graph continues to go down and up and to the left and right. So all of these numbers will be included as well. So we can see that the range is going to be all real numbers also. And for the cube root function, our x actually can be negative because we can see that the negative x's are included for this graph.
So those are some of the common functions that you'll see throughout this course and also in future math courses. So hopefully you found this helpful, and let me know if you have any questions.
