Problem 20 - A Graphical Approach to Precalculus with Limits

Question

If the \(x\) -intercept method leads to a horizontal line that coincides with the \(x\) -axis, what is the solution set of the equation? What special name is given to this kind of equation?

Step-by-Step Solution

Answer

The solution set is all real numbers \(x \in \mathbb{R}\), and the equation is called a zero function.

Step 1: Recognition of Horizontal Line Intercept

If a line is horizontal and coincides with the \(x\)-axis, it means the line runs along \(y = 0\). Since the line is horizontal and every point on the line has \(y = 0\), the entire graph of the equation is the \(x\)-axis.

Step 2: Identifying the Equation Form

A horizontal line that is the \(x\)-axis is represented by the equation \(y = 0\). This is a special case where the equation reduces to \(y = 0\), meaning every \(x\)-coordinate on the plane satisfies this equation.

Step 3: Understanding the \(x\)-Intercept

Since the line coincides with the \(x\)-axis, the \(x\)-intercept is present at every point on the \(x\)-axis, which means the line has infinitely many \(x\)-intercepts. Hence, there is essentially no limit for \(x\)-intercepts.

Step 4: Solution Set

The solution set for the equation of a line that coincides with the \(x\)-axis (\(y=0\)) is all real numbers for \(x\). This can be written as \(x \in \mathbb{R}\).

Step 5: Special Name for the Equation

The equation \(y = 0\) that represents a line coinciding with the \(x\)-axis is called a 'zero function' or a 'special case of a horizontal line'. This implies that its graph is a perfectly horizontal line without any slope.

Key Concepts

x-intercept

In mathematics, an "x-intercept" refers to the point where a graph intersects the x-axis of a coordinate plane. For a point to be an x-intercept, its y-coordinate must be equal to zero, because the x-axis itself is defined by the horizontal line where y = 0.

When trying to find the x-intercept of an equation, you set the y-component to zero and solve for x. In the specific case of a horizontal line that lies directly on the x-axis, like the one described in our problem, the x-intercept is indeed special. That line doesn't just cross the x-axis at a single point; it runs along the entire axis.

This means every point on the x-axis is an x-intercept for the equation y = 0. In other words, the x-intercepts are infinite, covering all possible real number values of x.

zero function

When dealing with functions, a "zero function" is particularly fascinating. It is a function whose output, regardless of the input, is always zero. Mathematically, this is expressed by the equation y = 0 where the value of y never changes as x varies.

In the context of our horizontal line problem, y = 0 signifies that the line rests on top of the x-axis, which is why it's called a zero function. This function is unique because its graph results in a straight, horizontal line coinciding with the x-axis, with no inclination or slope.

Characteristics of the zero function include:

• No x-dependence: The y-value remains constant at zero regardless of x.
• Direct horizontal presentation: Graphically represented by the entire x-axis.
• Infinitely many solutions for every x-coordinate.

solution set

In mathematics, the term "solution set" refers to the set of all solutions that satisfy a particular equation or inequality. When an equation describes a situation perfectly, any number or value that makes the equation true is part of the solution set.

For our horizontal line scenario, the equation that coincides with the x-axis is y = 0. The solution set for this equation is noteworthy. Since every point on the x-axis is a part of this graph, the solution set contains all possible values for x that are real numbers. Formally, this can be expressed as:

• \( ext{Solution set} = \{x \mid x \in \mathbb{R}\}\)
• Alternatively, simply as \( x \in \mathbb{R} \)

Here, \( \mathbb{R} \) is symbolic of the set of real numbers, identifying a range without limits, covering every real number possible.

real numbers

'Real numbers' form a very broad and inclusive set in mathematics, covering all numbers that can be found on the number line. This includes conventional integers like 0, 1, 2, fractions like 1/2, and irrational numbers such as \( \pi \) or \( \sqrt{2} \).

In terms of our horizontal line where y = 0, the solution set encompasses all real numbers for x, essentially signifying the line extends forever in both the positive and negative directions across the x-axis.

Some key points about real numbers:

• They include positive numbers, negative numbers, and zero.
• They cover both rational and irrational numbers.
• They cannot represent imaginary numbers or complex values.

Thus, when we say the solution set for y = 0 is all real numbers, it implies there is no restriction on the values x can take, as each satisfies the equation equally.