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. In this video, we're going to take a look at some of the common functions. These are the functions that will frequently show up throughout this course. It's definitely important to make sure we're familiar with these functions, 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 \( f(x) \) is equal to \( c \), and \( c \) can be any constant number. For example, if our function \( f(x) \) is equal to 2, this would be a constant function because 2 is just a constant. Notice on our graph that \( y = 2 \) is constantly the value in all directions. We can input any \( x \) value we want into this function. Thus, we would say that the domain goes from negative infinity to positive infinity, 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 from this function.
Next, let's take a look at the identity function. The identity function states that \( f(x) = x \). This function tells you that whatever you put into the function, you're going to get out of it. For example, if you put negative one in for \( x \), your output will be negative one. If you put 50 in for \( x \), you're going to get 50 as your output. Therefore, the domain and the range are all real numbers, since you can put any number into it and get any number out of it.
Now, let's take a look at the square function. The square function forms an interesting shape called a parabola. This parabola is a bowl-like shape that you see on the screen. The square function happens when \( f(x) = x^2 \). The domain for this function is all real numbers because all of the negative \( x \)'s and all of the positive \( x \)'s are defined by this curve, which continuously expands up and to the left and right. However, we do not include negative \( y \)'s in our range. Therefore, our range will go from 0 to positive infinity, including 0.
The cube function is represented by \( f(x) = x^3 \). For this graph, we include all the negative \( x \)'s and all the positive \( x \)'s. Therefore, the domain is all real numbers. The range also includes all negative and positive \( y \)'s, so the range is also all real numbers.
The square root function has the most restrictions of all the common functions. The graph continuously goes to the right and also goes up. We have all positive \( x \)'s, but no negative \( x \)'s are included. Thus, the domain goes from 0 (including 0) all the way to positive infinity. The range also goes from 0 to positive infinity, as no negative \( y \) values are included.
The cube root function, represented by \( f(x) = \sqrt[3]{x} \), includes all negative and positive \( x \)'s. Therefore, the domain goes from negative infinity to positive infinity, including all real numbers. The range also includes all negative and positive \( y \)'s, so it is all real numbers too.
These are some of the common functions that you will see throughout this course and also in future math courses. Hopefully, you found this helpful, and let me know if you have any questions.