## What Happens When We Reduce Fractions to Lowest Terms

Many mathematical algorithms are performed so routinely that we sometimes do not think about why we do them or why the method works. One example of such routine algorithm is reducing fractions to lowest terms. Why do we always perform this procedure? What is really happening when we reduce fraction to lowest terms?

One of the basic reasons why we reduce fraction to lowest terms is to lessen the burden of calculating large numbers. Of course, we would rather add or multiply $\frac{1}{2}$ and $\frac{1}{4}$ than $\frac{12}{24}$ and $\frac{9}{36}$. So you see, the effort of multiplying the same fraction is lessen when they are reduced to lowest terms.

But what really happens when we reduce fraction to lowest terms? Why is it possible and why does it work? » Read more

## Domain and Range on a Graphical Perspective

Two weeks ago, I  discussed the basic concepts of domain and range which I presented in an ‘algebraic way.’ In this post, I would like to discuss these concepts from a graphical perspective.

The domain of a function $x$ is the set of points on the x-axis where if a vertical line is drawn, it will hit a point on the graph. Take for instance, in the linear function $f(x) = 2x$,  we are sure that we can always hit a point wherever we draw a vertical line. In algebraic explanation, we can always find an $f(x)$ for every $x$. Therefore, we can conclude the that domain of $f$ is the set of real numbers. On the other hand, if we draw a horizontal line and it hits the graph, then it is part of the range of the graph. Clearly, the range of the $f$ is also the set of real numbers.

## Domain and Range 1: Basic Concepts

Domain and range are concepts that are essential in learning functions.  In most resources, these concepts are just defined technically, and although there are examples, many just lack intuitive explanations. In this post, we discuss domain and range in a simple and hopefully easy to understand manner.

Example 1: $f(x) = 2x + 1$

Domain

What is the domain of $f(x) = 2x + 1$ and what is its range? Well, the domain are just the possible values of $x$ that will produce a “valid” value $f(x)$. To check, we can ask the following questions.

• Can we substitute positive values to $x$?
• Can we substitute negative values to $x$?
• Can we substitute 0 to $x$?

Obviously, the answer to these questions are all yes. In fact, we can assign any real number value to $x$ and we can always get a corresponding value for $f(x)$. » Read more

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