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One of my previous posts introduced you to functions and how they work, but there are few other things worth discussing that I would like to add here. So, now that you know how to differentiate between and name different functions, we can do slightly more complicated things – things like complex functions. The functions that I’ve discussed so far (and generally, what most people typically talk about) can be drawn as continuous, smooth curves with no gaps. The notation basically says that here is the function f(x), and it is equal to the listed equations over the specified domains. With that, I will conclude my brief introduction to complex functions and piecewise functions, and I will look ahead to my next post where I will go into a bit more detail about graphing piecewise functions and some things to keep in mind, as I have hinted at above. We present an introduction and the definition of the concept of continuous functions in calculus with examples. In the graphs below, the limits of the function to the left and to the right are not equal and therefore the limit at x = 3 does not exist. The limits of the function at x = 2 exists but it is not equal to the value of the function at x = 2. The limits of the function at x = 3 does does not exist since to the left and to the right of 3 the function either increases or decreases undefinitely.

The first is more about function notations, then I will expand that into a demonstration of complex functions, and then wrap up this post by introducing the concept of piecewise functions. However, you will quickly come across problems that use different letters – this means the same thing. Continuing our metaphor, it’s like saying that our function machine f(x) has an input of the entire machine g(x). We can evaluate the expression as we just did, then solve when x= 5 (like I said above, this is like giving the f(x) machine the entire g(x) machine as its input. I’ll post a follow-up shortly to focus on graphing piecewise functions, but here is an introduction to them.

The functions whose graphs are shown below are said to be continuous since these graphs have no "breaks", "gaps" or "holes".

You will probably see things like g(x) or h(x), and they are just used to differentiate between different functions. The f, g, and h are all essentially saying the same thing – that is, they are the names of their function. Function notation can get very confusing like this, but for problems like this, you just have to basically work from the inside out, just as in any other math expression you’ve seen!

But what if you consider a graph that is composed of two distinct sections – such as one with a rising diagonal line attached to a horizontal line at the top? Both methods will get you the same answer, and it’s really up to you to do whichever you are most comfortable with (unless you are instructed to specifically do one way or the other!). In this case, at the point on the curve corresponding to x = 2.5, both equations are valid and will produce the same value. However, combining these sections into graphs such as these, which are described by multiple equations each applying to specific domains of the entire curve, these are called piecewise functions.

They may seem tricky, but you will be surprised at how easy they are when you see through the notation and know what they are asking!

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