Unit 3: Functions

Section 3.1: The definition of a function.

Video: The definition of function. Domain and Range.

Exercise set 3.1A

  • p.140: #5-8 all 

Optional Video: Intro/Review of Function notation; Machine Metaphor.

Exercise set 3.1B

  • p.140#1-4 all (if this goes quickly, that is great, if not, be grateful for the practice).

Video: Domain and range intuition.

Exercise set 3.1C

  • p.143 #60 a,b,c. Most importantly: Describe what a fixed point is, using the words input and output. Draw a function machine diagram explaining the concept. Now sketch and use the graph of one of the equations from part a,b,or c to explain the concept.

Video: Determining domain and range algebraically.

Extra resources for domain and range:

The following Khan academy videos are a pretty good idea if you just want to see a variety of extra examples of finding the domains and ranges of functions. He covers several situations in which the domains and ranges are interesting sets that are not just continuous subsets of the real numbers. For example, sets that are multiples of a ticket price.

  • This Khan video is helpful for developing some conceptual understanding for why we need the domain and range concepts.
  • This Khan video is more of the kinds of examples from my video above. That is, more algebraic work for determining the domain of a function.
  • This Khan video shows several more Domain and Range examples.

Exercise set 3.1D

  • p.141 #9-25 odd
  • p.142#29, part a,b,c,f,g,h,k
  • Carefully read from about the middle of page 136 (“We often use single letters…”) through to the end of the chapter. The goal is to understand function notation such that problems like Example 8 on page 140 make sense. Also, make sure that you have some idea of what a demand function is.

Video: Demand Functions

Exercise set 3.1E

  • p.142, #30, part a.) only;  #31 (all parts) <-These problems are essentially about function notation.
  • p.142 #39, 41, 42 <-These are about demand functions.
  • p.142 # 45-48 <- These problems show you that certain tempting operations do not work with function inputs.
  • p.142 #58 (all parts) <- Functions are cool, partly because their definitions can be pretty abstract. Turns out this function plays a key role in detecting election fraud, as well as all sorts of other kinds of fraud. Oddly, it all began with the smudges in a huge book of numbers (called logarithms).


  • p. 143Write programs that solves #63 for any input x.
  • p. 143 Write a program that solves #64 a given number of times. Prompt the user for the number of values of x for which P(x) is not prime and the program should find and print that many values. It should be easy to do the same thing for part b. after you have solved it for part a.
  • p. 143, # 66

Section 3.2: The graph of a function.

 Video: The technical definition of the graph of a function and the vertical line test.

Exercise set 3.2A

  • p.153, #1-6 all
  • p.154 #7-14 <-These should go quickly once you get the hang of it. See example 3 and 4 from the section (starting on page 147).
  • p.155 #17,18,19,20
  • p.156, #33,35


  • p.156, #36
  • p.156, #38


 Video: Graphing piecewise functions.

Exercise set 3.2B

  • p.155, #23-29 odd


  • Pick three of the exercises from the project on page 157.


Section 3.3: The anatomy of graphs and average rate of change.


 Video: Graph anatomy. Asymptotes.

 Video: Graph anatomy. Increasing and decreasing.



Video: Maxima and minima.

Video: The average rate of change of a function.

Exercise set 3.3

  • p. 166. Complete questions 1 and 2 and discuss them with a classmate.
  • p.167, #3-7 all
  • p.168, #9,13,15,17,18 (let me know if you need to make a copy)
  • p.169, #19b, 21,23,24,25,27,31,33,35
  • p.70, #37, 41,43


  • I have changed this assignment slightly. Get the new version here: finding_e_with_bisection.
  • This assignment is pretty involved so I plan to make this due when the work for 3.5 is due.

Section 3.4: Techniques in graphing.

By now you are probably familiar with the graphs of several functions. On page 149 in section 3.2, you encountered 6 important graphs. They were so important that the authors encouraged you to memorize them.

Do so, if you have not already.

You might be wondering what the point is in memorizing these 6 graphs. Surely, while attempting to model some system in your future as a math-wielding thing-doer, surely you will encounter countless systems that can not be modeled by one of these 6 graphs. There is an infinite number of possible graphs. Sure, you know the graph of y=|x|, but what about y = |x-1|. And if you memorize that, what about y = |x-2| and so on and so on?

It turns out that if you know the graph of y=|x|, then it is easy to graph these other equations. The six graphs that you memorized can be easily modified to cover an infinite number of other cases, that is, to create an infinite number of other graphs. After a little practice, these 6 graphs will bend to your will, shifting and reflecting about on the coordinate plane like so many… um… shifted and reflected graphs.

Video: Reflections.

Video: Translations.

Video: Translation and Reflection Examples

When a function is changed by taking the negative of the independent variable (the input variable), the process for graphing this new function is not obvious. Luckily, it turns out that obtaining the graph is pretty straight forward. The reasoning behind the procedure is a little involved. This video explains “the why and how” with a concrete example.

Exercise set 3.4

  • p. 178, #1
  • p.179, #2,3-39 ever other odd, also 16 and 24.
  • p.180, #54,55,59,61 a.)


Section 3.5: Function operators and Iteration.

When I first started teaching Precalculus, about 8 years ago, I thought that the operations on functions section was pretty useless and irrelevant. Then I studied abstract algebra.  The notion of operations on a set (such as the set of real-valued functions… yes, that is a valid set!) forms the cornerstone of several branches of mathematics. One of these branches is ridiculously awesome (as in borders on sorcery) and among other things explains how old-timey people were able to multiply giant numbers by adding very small ones, how to come up with little theories, like say, modern quantum mechanics and mathematically un-crackable encryption algorithms (well, at least until we all get quantum computers).


This section does not go into group theory, although you do learn about some binary operations on the set of all real valued functions. We just won’t call them that. If you are wondering what all of this preamble was about, you might start with this article. Some of the article will not make sense so feel free to ask if you have any questions.

Optional Video: What is an operation, anyway?

Video: Arethmatic operations on functions

Video: Function composition, intuition and examples.

Video: Definition of function composition

After watching the following video, be sure to carefully read Example 4 in section 3.5 (I am looking at the 5th edition, so it might be a different example. Maybe ask me about this?)

Video: Function composition examples.

Video: Function composition modeling problem (example).

Video: Iteration


Exercise set 3.5

  • p. 189 #1,3
  • p. 190#9-11,14,17,19,
  • p.191 #21,25,27,29
  • p. 192 #31,35,41


  • Write a program that tests the 3x+1 conjecture for 0 < x 0 < 100. See problem 43 on pages 192-193 for more information about this conjecture.
  • p. 194 “Mini Project”.

Section 3.6: Inverse Functions.


Video: Inverse Functions: Intuition and Definition



Video: Inverse Functions: Examples (including finding a rule for the inverse of a functino)

Video: Inverse Functions: The graph of the inverse of a function (reflection in y=x)

Video: One-to-one intuition.

Video: Definition of One-to-one and the Horizontal line test.


Exercise set 3.6

  • p.203, #1-5 all,7-9 all
  • p.204, #11,13,15,17,21,23,25,27,29,31,33,34
  • Practice Test / Review Exercises

  • p.209, # 1,2,5,7,16 (optional p. 204, #26 for extra practice),28,29,39,54,55,81-84,90-91
  • p.177, Do example 7 without looking at the solution (it is an example problem in the chapter).
  • p.204, #24
  • p.192, #42