## What exactly is the vertical line test?

A function as we have discussed is a relationship between two sets, where each element in the first set has exactly one corresponding element in the second set. If we think of candies which cost 10 cents each, then we can say that 1 candy costs 10 cents, 2 candies cost 20 cents, 3 candies cost 30 cents, and so on. We can think of this relationship as a function since for each number of candies, there is only one possible price.

If we consider the relation y = 2x, then we can say that it is a function since for every value we substitute to x, there is one and only one corresponding value for y. For instance, if x = -3, then y = -6 and and if x = 9, then y = 18 (one y for each x).  » 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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