Beginning with your earliest experiences with plotting functions, you have intuitively plotted the sum of two functions. Consider, for example \(h(x)=3x-x^2\text{.}\) You can think of \(h(x)\) as the sum of two simpler functions, \(f(x)=3x\) and \(g(x)=-x^2\text{.}\) In this section, we will carefully explore what it means geometrically to add two simple functions together. Then we will apply this geometric interpretation to some special cases that are important in understanding Chapter 13 Power Series (Power Series), and Chapter 20 (Fourier series).
ActivityA.2.Visualizing the Pointwise Addition of Functions.
In the applet below, you can enter any two functions of your choice, \(f(x)\) and \(g(x)\text{.}\) (The default choices are \(f(x)=3x\) and \(g(x)=-x^2\text{.}\) With the default check boxes clicked, you will immediately see graphs of these two functions.)
First, try to sketch by hand the sum of these two functions \(h(x)=f(x)+g(x)\text{.}\) Describe your process for making the sketch. Are there any features of the graphs that you pay attention to when you make your sketch?
Now click the third check box so that the applet shows you the graph of \(h(x)\text{.}\) What features of your sketch were correct? Add anything you discover to your list of features to pay attention to.
FigureA.5.In this interactive applet, you can enter two functions of your choice. The default functions displayed are \(f(x)=3x\) and \(g(x)=x^2\text{.}\) A check box reveals the sum \(h(x)=f(x)+g(x)\text{.}\) You can drag the graph to center the region that you care about and use the scroll wheel or pinch to zoom in. A full-screen version of this applet can be found at GMM: Adding Functions Pointwise 1
If you think of \(x\) as representing a specific number, then \(f(x)\text{,}\)\(g(x)\text{,}\) and \(h(x)\) are all numbers and addition just means addition of numbers. On the graph, find the values of \(f(x)\) and \(g(x)\) and add them together as numbers to find the value of \(h(x)\text{.}\) In complicated cases, it can be helpful to make a table of values, including columns for \(x\text{,}\)\(f(x)\text{,}\)\(g(x)\text{,}\) and \(h(x)\text{.}\)
In your sketch, you might want to pay attention to where the input functions are zero, positive, negative, larger or smaller than each other, increasing or decreasing.
DefinitionA.6.Pointwise Addition of Functions.
The strategy of adding two functions together by thinking of specific values of the independent variable (e.g. \(x\)) and adding together the numbers\(f(x)\) and \(g(x)\) is called pointwise addition of functions.
There is nothing mysterious going on here. Pointwise addition is what you have always done when plotting functions. But in more complicated situations such as the definition of power series (13.1.1) or Fourier series (20.2.1), where you may be adding up an infinite number of terms and where you are likely to be paying attention to each of the individual terms as you add up more and more, it can be easy to lose sight of the geometric meaning of what you are doing.
Important Examples.
In power series, the terms you are adding are powers of the independent variable and you care most about small values of this variable. It is important to know which powers make a larger contribution to the sum. To explore this idea, use the applet above with \(f(x)=10x\) and \(f(x)=x^2\text{.}\) Which term is larger as \(x\rightarrow 0\text{?}\) Make sure to scroll drag the graph as necessary. The answer to this question is deceptive.
Similarly, in Laurent series, the terms you are adding are inverse powers of the independent variable and you care most about large values of this variable. It is important to know which inverse powers make a larger contribution to the sum. To explore this idea, use the applet above with \(f(x)=10/x\) and \(g(x)=1/x^2\text{.}\) Which term is larger as \(x\rightarrow 0\text{?}\) Make sure to scroll drag the graph as necessary. The answer to this question is deceptive.
In Fourier series, the terms you are adding up are sines and cosines whose periods are integer multiples of each other. In this case, large terms are obvious, they have large coefficients. Instead, what your want to notice is that the sum of two of these functions is still a periodic function. To explore this idea, use the applet above with \(f(x)=\sin mx\) and \(g(x)=\sin m'x\) or \(g(x)=\cos m'x\text{.}\) What is the period of the sum function \(h(x)\text{?}\)
