The next topic to review is that of composite functions. I feel that I understand the concept but I could use a little practice by working through a few of the examples in the Banner book.

• The composition of two functions, f o g is obtained by substituting the expression for g for each occurrence of x in the expression for f, as long as the resulting expression still makes sense and is defined for all x in the domain of f.
  
  
• "Its useful to practice composing two or more functions together" [p. 12] ... and to also practice the reverse process where one begins with a complex expression and tries to reduce it to a sequence of nested simple functions.

I had no difficulty doing the two examples in the text. But the real question is why. When would one want to do this, and for what reason? I think the answer is when one wants to understand the behavior of a complex expression by breaking it down into a series of simpler expressions. I want to try this with the two examples I have just worked out by using Mathematica to give me the graphs of the intermediate expressions. Now I need to have a look at Mathematica. I recently bought a book on Mathematica and I will have a look at this when I return to Lethbridge.

• "... a function is even if f(-x) = f(x) for all x in the domain of f ... the graph of an even function has mirror symmetry about the y-axis." [p. 14 - 15]
  
  
• "... a function is odd if f(-x) = -f(x) for all x in the domain of f ... the graph of an odd function has 180 degree point symmetry about the origin." [p. 14 -15]
  
  
• "In general. a function might be odd, it might be even, or it might be neither odd nor even. ... Most functions are neither odd nor even." [p. 14]
  
  
• "One nice thing about knowing if a function is odd or even is that it is easier to graph the function." [p. 14]
  
  
• "Functions of the form f(x) = mx + b are called linear. There's a good reason for this: the graphs of these functions are lines." [p. 17]
  
  
• "To sketch the graph of a linear function, you only need to identify two points on the graph." [p. 17]
  
  
• "Since the functions of x raised to the power n are the building blocks of all polynomials, you should know what their graphs look like." [p. 19]
  
  
• The even powers mostly look similar to each other (i.e. parabolas) and the same can be said for the odd powers (i.e. curvy).
  
  
• "Sketching the graphs of more general polynomials is more difficult. Even finding the x-intercepts is often impossible unless the polynomial is very simple. There is one aspect of the graph that is fairly straightforward, which is what happens at the far left and right sides of the graph. This is determined by the so-called leading coefficient, which is the coefficient of the highest degree term. It only matters whether the leading coefficient is positive or negative and whether the degree of the polynomial is odd or even, so there are four possibilities. ... The wiggles in the center of the graph are dependent on the other terms and are not relevant for determining what happens at the far edges of the graph." [p. 20]

I need to commit the general shape of these four situations to memory!

• "The general shape of a rational function p(x) / q(x) also has some general features, at least when the numerator is 1. Graphs when q is an odd power function all look similar and so do graphs when q is an even power function.

I really should spend some time looking at the shapes of some of these simple polynomial functions. Chapter 2 of Banner is all about a review of trigonometry. This is also worth spending some time on as I have forgotten most of the basic relationships. Tomorrow.

9:30 am

Tags: mathematics, calculus