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Chapter 3: Problem 21

Decide whether the equation describes a function. $$y=6$$

Short Answer

Expert verified
Yes, the equation describes a function.

Step by step solution

01

Identify the Equation Type

The given equation is a horizontal line equation: \( y = 6 \). This means that no matter what value we choose for \( x \), the value of \( y \) remains constant at 6.
02

Determine the Output for Each Input

For a function to be valid, each input should have exactly one output. In this case, regardless of what \( x \) value we select, the output \( y \) is always 6. So, for any input \( x \), there is exactly one corresponding output \( y = 6 \).
03

Check the Vertical Line Test

A graphical check can help verify if an equation is a function. According to the vertical line test, if a vertical line can intersect the graph at more than one point, then the relation is not a function. However, since this graph is a horizontal line, any vertical line will intersect it at exactly one point, confirming that it is a function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Line Equation
In algebra, a horizontal line equation is known for its simplicity. It is expressed in the form \( y = c \). Here, \( c \) is a constant number. For example, the equation \( y = 6 \) means that the value of \( y \) is always 6, no matter the value chosen for \( x \). This is because a horizontal line runs parallel to the x-axis.

Key characteristics of a horizontal line include:
  • The slope of the line is zero since it doesn鈥檛 rise or fall as it moves horizontally.
  • The line goes straight across at a fixed y-value.
  • Every point on the line has the form \( (x, c) \).
Exploring further, consider plotting \( y = 6 \) on a graph. The line cuts through the y-axis at 6 and runs parallel to the x-axis indefinitely.

This tells us that horizontal equations are incredibly straightforward, providing a clear and constant output no matter the x-value.
Vertical Line Test
The vertical line test is a simple visual method to determine if a graph represents a function. By this test, a relation is a function if and only if no vertical line can cross the graph more than once. This is an easy way to tell if each input has a single output.

Let's break it down:
  • Draw or imagine vertical lines (up-down) through different parts of the graph.
  • If a line touches the graph at more than one point, it's not a function.
  • If it only touches the graph once per line, it is a function.
Applying this to our equation \( y = 6 \), the graph is a horizontal line at y = 6. Any vertical line you draw will intersect the graph at exactly one point. This confirms that \( y = 6 \) is indeed a function because each x-coordinate is associated with exactly one y-value.
Constant Function
A constant function is a unique case in mathematics where the output or result remains unchanged regardless of the input. This means that no matter what x-value you plug into the function, the y-value remains the same.

Using our example of \( y = 6 \):
  • This is a perfect example of a constant function because no matter the x, y is unchanging at 6.
  • The graph of \( y = 6 \) is a flat horizontal line, showing visually that y does not depend on x.
  • Constant functions are therefore perfectly predictable and consistent.
Constant functions are crucial in various mathematical applications due to their predictable nature. They are often considered the simplest type of function, making them an important foundational concept in algebra.

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Most popular questions from this chapter

Solve. See the Concept Check in this section. In your own words, describe how to plot or graph an ordered pair of numbers.

Solve each equation for \(y .\) See Section 2.5 \(4 y=-8 x\)

Professional plumbers suggest that a sewer pipe should rise 0.25 inch for every horizontal foot. Find the recommended slope for a sewer pipe. Round to the nearest hundredth.

Determine whether each pair of lines is parallel, perpendicular, or neither. \(y=\frac{1}{5} x+20\) \(y=-\frac{1}{5} x\)

Solve each equation for \(y .\) See Section 2.5 \(x-3 y=6\)
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