From an early age, we are taught about functions without any real understanding of what they are. This lack of understanding eventually causes problems when a need for more advanced functions arises. This series will cover what schools fail to teach: understanding. It is my hope that you will not simply memorize the things that I write, but instead look closer at what I mean. All graphs are done on Desmos.
Part 1: What is a function?
Simply put, A function is any operation which maps an input to a single output.
This definition, though, is utterly useless without first understanding what it means.
Say I have some point, p, on the number line. Points on the number line represent numbers, the same way that variables like x can. Our function will take this point p and move it. It cannot make new points, which is where the part about the single output comes in.
You can think of a function as a road. The point p can travel down that road to get to another point, but traveling down that road doesn't make a new point spring into existence. Roads have names, so let’s call this function h. It can be said that h(p) (or h of p) is equal to some other point---which we’ll call q. This means that h(p)=q. As shown below, you can see that all function h does is move point p to the location of point q (Figure 1.1).
Congratulations, you have now used a function!
This function isn't very useful, however. For one thing, it only has a single input, which means it can only act on a single point at once. We want more than one input, preferably infinite (Figure 1.2). Rather than assign each h(p)=q for an infinite number of points, we use a shorthand. We say that h(x)=y, where y is an equation and x represents the entire input space at once. For example, if we say that h(x)=x*2, we say that for any x, h(x) is x*2. Note that the following only graph certain key points.
Now try h(x)=x² (Figure 1.3).
If we try to graph this on a number line, it doesn't look very good. This is because more than one input can map to the same output. This is akin to two people walking down a road to the same place. Maybe we need to find a new way to graph these things, so that the input space doesn’t overlap with the output space. Let’s try putting the output space normal (i.e. perpendicular) to the input space (Figure 1.4).
This is much better. Notice that this allows us to graph continuously, as opposed to only a few points at a time. This representation is called the Cartesian Plane. It gives the position of a point using two coordinates: the x-coordinate, which gives the left-right position, and the y coordinate, which gives the up-down position.
Though it may appear to seem like the number line was simply bent to form that shape, this isn’t the case. Imagine a set of blocks which can move up and down only. In this mapping, the x-coordinate of each point doesn't change. All curves are simply the points being 'pushed' up the perfect amount so that a curve appears.
It's important to realize the difference between this graph and what you learned in school. This graph is the number line being pushed upward. The line has to stretch, meaning that the distance between points increases, but only in the vertical direction. Here, those of you who have taken Calculus can see that the amount that the line stretches is the derivative, but the amount that the line is pushed is the integral (Figure 1.5).
Part 2: Translation of functions
Often we are taught about the basic translations of functions like so (Figure 2.1, 2.2):
Let’s now look at the intuitions behind them.
f(x)+1 takes every point that is shifted up on the number line, and is shifted up by one more (Figure 2.1). If f(x)=y, f(x)+k=y+k.
The other transformation is a little more difficult to grasp (Figure 2.2). Think about how a function just moves these points, without changing the x position. These functions now raise the point to where the point at x+k should be raised. If you remove the x, there is no distinction between this new point and the same point on the graph of f(x). This makes it seem like the graph is moving left, when in actuality it is moving up and down in such a way that it looks like it is moving left (think of a waving jump rope. It doesn't move horizontally, but the wave looks like it does).
Part 3: Functions of functions
I would like you to look at the following functions:
Now when asked what g(f(x)) is, you may recall what school taught you to do:
While this is very accurate and quite easy to do, it gives no sense of visualization. What, you may wonder, does such a graph look like?
Here we must refer to the previous visualization of each point on a number line mapping to another. We first graph g(x) normally:
Now move the number line, or x coordinates, to f(x).
The resulting graph is g(f(x)). This is similar to a horizontal translation of every point on g(x) to the output space of f(x).
Note to reader: this post will be continuously updated with more parts. Please check back soon for Part Four: Functions of Other Inputs