What is the horizontal line test?

In January of this year, we have discussed about the vertical line test. We have learned that if a vertical line intersect a graph more than once, then that graph is not a function. In this post, we learn about the horizontal line test and its relation to inverse functions.

Suppose we have a function f. Then, we input x and call the output f(x). If we do things backward, suppose we have the number y which is in the range of f. What value should we input to f to get y? Let’s have a more specific example.

Suppose we have the function f(x) = x^2. And we choose the number y = 9 which is in the range of f. What number should we input in f to get 9? Well, we will have two numbers, those are 3 and -3. » Read more

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

Basic Concepts of Functions

Note:  This is the second part of the Functions Series. To view the other parts, click the link below.

Part I: Introduction to Functions
Part II: Basic Concepts of Functions


In the first part of this series, we have discussed that a function is a relationship between two sets where for each value in the first set, there is exactly one corresponding value in the second set. We have painted large cubes, cut them into unit cubes and found a pattern about the number of cubes with 3, 2, 1 and no painted faces.

Figure 1 – Cubes painted and sliced into unit cubes.

We found out that if a cube has side n units, if we painted all of its faces and cut it into unit cubes, the following relationships hold: » Read more

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