Problem 112 - Algebra for College Students

Question

Consider the graphs of \(y=2\) and \(x=3\) in the rectangular coordinate system. Explain why one of these relations is a function and the other is not.

Step-by-Step Solution

Answer

The relation \(y=2\) is a function because each x-value maps to one y-value. The relation \(x=3\) is not a function because one x-value maps to multiple y-values.

Step 1 - Understanding Functions

A function from a set of inputs to a set of possible outputs is defined such that each input is related to exactly one output. This means that in a function, no input can map to multiple outputs.

Step 2 - Analyze the Graph of y=2

The graph of the equation \(y=2\) represents a horizontal line crossing the y-axis at 2. For any value of x, the output is always 2. Hence, each x-value maps to only one y-value.

Step 3 - Determine if y=2 is a Function

Since for every x-coordinate, there is a unique y-coordinate (which is 2), \(y=2\) satisfies the definition of a function.

Step 4 - Analyze the Graph of x=3

The graph of the equation \(x=3\) represents a vertical line crossing the x-axis at 3. For any value of y, the value of x is always 3.

Step 5 - Determine if x=3 is a Function

According to the definition, a function cannot have the same input mapping to multiple outputs. Since \(x=3\) has the same x-value (3) for any y-value, it violates this rule, hence it is not a function.

Key Concepts

rectangular coordinate system

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane for graphing equations. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.
Each point in this system is represented by an ordered pair \( (x, y) \). Here, 'x' is the coordinate along the x-axis, while 'y' is the coordinate along the y-axis.
This system makes it easier to visualize mathematical concepts and relationships by plotting them on a graph. Understanding the rectangular coordinate system is essential for studying algebra and functions.

definition of a function

In algebra, a function is a special relationship between two sets: a set of inputs (domain) and a set of possible outputs (range). A function assigns each input exactly one output.
To determine if a relationship is a function, you can use the 'vertical line test'. Draw a vertical line through the graph of the relationship:

• If the line intersects the graph at more than one point, the relationship is not a function.
• If the line intersects at exactly one point, the relationship is a function.

For example, the equation \( y=2 \) (a horizontal line) passes the vertical line test, confirming it is a function. However, the equation \( x=3 \) (a vertical line) fails this test since it intersects the graph at multiple points, showing it is not a function.

horizontal and vertical lines

Horizontal and vertical lines are special types of lines in the rectangular coordinate system.
A horizontal line has a constant y-value and takes the form \( y=b \), where 'b' is a constant. This line runs parallel to the x-axis and crosses the y-axis at the point \( (0, b) \). An example is \( y=2 \), which means every point on the line has a y-coordinate of 2.
A vertical line, on the other hand, has a constant x-value and is written as \( x=a \), where 'a' is a constant. This line runs parallel to the y-axis and crosses the x-axis at the point \( (a, 0) \). An example is \( x=3 \), indicating every point on the line has an x-coordinate of 3.
When analyzing these lines using the vertical line test:

• Horizontal lines are functions because each x-value maps to a single y-value.
• Vertical lines are not functions because a single x-value maps to multiple y-values.

Understanding these properties helps in determining whether different types of lines are functions.