# Book of Proof: A Very Good Introductory Book to Mathematical Proofs

Two weeks ago, I finished reading Book of Proof  (link goes to Amazon) by Professor Richard Hammack, and so far, it was the best book that I have read about introduction to mathematical proofs. I recommend this book to high school students who are interested in pursuing a mathematics degree, to college students who are math majors, and to teachers who will teach or who are teaching an introductory course on mathematical proofs. Here are some of the reasons why I really like this book.

Clear Explanations

The explanations of concepts in this book are very clear and the concepts are well-connected. Maybe this is because the author’s long experience in teaching this course (or maybe he is just a very good teacher). This book is a product of the author’s lecture notes on teaching mathematical proofs for the past 14 years.  Continue reading

# Free Peer-Reviewed Math Ebooks from OpenStax

If you are looking for free math resources, I found a website that offers peer-reviewed math ebooks for college and AP Courses. This website is OpenStax which is based in Rice University and supported by several foundations. Below are the links to the math ebooks.

Aside from the math ebooks on Science, Social Science, Humanities, and AP Courses are available on the website.

For more free resources, you can visit Math and Multimedia’s All for Free page.

# Understanding Domain and Range Part 4

In this post, we summarize the previous three articles about domain and range. In the first part of the series, we focused on the graphical meaning of domain and range. We have learned that the domain of a function can be interpreted as the projection of its graph to the x-axis. Similarly, the range of the function is the projection of its graph to the y-axis.

Graphical meaning of domain (red) and range (green)

In the second part of the series, we learned to analyze equations of functions to determine their domain and range. We learned the restrictions in the domain and range of functions are affected by the following: squares in the expressions, square root signs, absolute value signs, and being in the denominator. In exploring these we concluded the following:

• Expressions under the square root sign result to a positive real number or 0.  This means that we have to set the inequality such that the expression is greater than or equal to 0, and then find the permissible values of x.
• Expressions containing squares result to a positive real number or 0. This affects the range of the function.
• Expressions inside the absolute value sign result to a positive number of 0. This also affects the range of the function.
• Expressions in the denominator of fractions cannot be 0 because it will make the function undefined. So, we need to find the value of x that makes the denominator by 0. To do this, we equate the expression in the denominator to 0 and find the value of x. The values of x are the restrictions in the domain.

In the third part of the series, we examined functions that have more complicated equations than those in the second part of the series.

Before I end this series, there is one more concept about domain that I want you to remember. That is, the domain of all polynomial functions is the set of real numbers. That’s why the domain of linear functions and quadratic functions in Part 1 and Part 2 is the set of real numbers.

# Understanding Domain and Range Part 3

In the previous post, we have learned how to analyze equations of functions and determine their domain and range. We have observed that the range of the functions $y = x^2$ and $y = |x|$ are the set of real numbers greater than or equal to $0$ since squaring a number or getting its absolute value results to $0$ or a positive real number. We also learned that for a function to be defined,  the number under the square root sign must be greater than or equal to 0. Lastly, we have learned that we cannot divide by zero because it will make the function undefined.

In this post, we are going to continue our discussion by examining functions with equations more complicated than those in the second part of this series.

Squares and Absolute Values

1. $f(x) = x^2 - 3$

Domain: The function is defined for any real number $x$, so the domain of $f$ is the set of real numbers.

Range: The minimum value of $x^2$ is $0$ for any real number $x$ and $f(0) - 3 = 0^2 - 3 = -3$. So, the minimum value of the function is $-3$. We can make the value of the function as large as possible by increasing the absolute value of $x$. So, the range of the function is the set of real numbers greater than or equal to $-3$ or $[-3, \infty)$ in interval notation.  Continue reading

# Understanding Domain and Range Part 2

In the previous post, we have learned the graphical representation of domain and range. The domain of the function $f$ is the shadow or projection of the graph of $f$ to the x-axis (see the red segment in the figure below). The range of $f$ is the projection of the graph of $f$ to the y-axis (see the green segment in the figure below). In this post, we are going to learn how to analyze equations of functions and determine their domain and range without graphing.

If a graph of a function is projected to the x-axis, the projection is the set of x-coordinates of the graph. A single point $(a,0)$ on the projection means a point on the graph exists. The existence of a point implies that $f(a)$ exists. This means that the function is defined at $x = a$. In effect, the domain of a function is the set of x-coordinates that makes the function defined. In what follows, we learn some examples to illustrate this concept.  Continue reading