Problem 72 - College Algebra

Question

What is the horizontal line test and what does it indicate?

Step-by-Step Solution

Answer

The horizontal line test is a method used to determine if a function is injective (one-to-one). This is done by drawing a horizontal line through the graph of the function. If the line intersects the graph at more than one point, the function is not injective. The test indicates whether every input to the function will result in a unique output.

Step 1: Definition of the Horizontal Line Test

The Horizontal Line Test is a method used in mathematics to determine whether a function is injective or not. A function is said to be injective if every element of the function's domain maps to a unique element of its range.

Step 2: Application of the Horizontal Line Test

To apply the horizontal line test, a horizontal line is drawn through the graph of the function. If at any point the horizontal line intersects the graph at more than one point, the function is not injective. In other words, if it is possible for the horizontal line to touch the graph in more than one place, the function has more than one y value for at least one x value, which contradicts the definition of an injective function.

Step 3: Indication of the Horizontal Line Test

The Horizontal Line Test effectively tells us whether a function is injective or not. If a function passes the Horizontal Line Test, it means that the function is injective, and every input to the function will result in a unique output. If a function fails the Horizontal Line Test, it is not injective, indicating that there is an input for which there are multiple outputs.