Problem 58 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 \pm 4 i}{2}=1 \pm 4 i$$

Step-by-Step Solution

Answer

The statement is false. A corrected version of the statement would be: The graph of every line, except vertical lines, is a function.

Step 1: Define Function

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Step 2: Understand Line as Function

In a Cartesian coordinate system, a line can be represented as a function if for every x-value there is exactly one y-value. This is commonly known as the vertical line test.

Step 3: Evaluating the Statement

So looking at the statement 'The graph of every line is a function.' we see that this is generally true for lines that are not vertical. However, vertical lines do not meet the requirements of a function because there are many y-values for the same x-value. Therefore, the initial statement is false.

Step 4: Correcting the Statement

To correct the given false statement, we could say: 'The graph of every line, except vertical lines, is a function.'

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system is a foundational element in algebra and graphing. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. Together, these axes divide the plane into four quadrants and allow us to plot points, lines, and curves. This system is incredibly useful in understanding the graphical representation of equations. A point in this system is represented as an ordered pair \( (x, y) \), where \( x \) is the position on the horizontal axis, and \( y \) is the position on the vertical axis.

• Visualization: Points and lines can easily be drawn, helping provide visual clarity on equations.
• Representation: Equations like \( y = mx + b \) for linear functions can be clearly showcased.
• Analysis: Physical movements and relationships between quantities can be better understood.

vertical line test

The vertical line test is an essential tool in determining whether a graph represents a function. To use this test, imagine drawing a vertical line (a line parallel to the y-axis) through any point on the graph.

• If the line intersects the graph at more than one point, then the graph is not a representation of a function. This is because a function must have only one unique y-value for each x-value.
• Conversely, if each vertical line touches the graph at only one point, then the graph is that of a function.

A line in the Cartesian coordinate system is typically a function, except for vertical lines, as they fail this test by intersecting multiple y-values at a singular x-value. Thus, the statement "The graph of every line is a function" is false because vertical lines do not qualify as functions under this criterion.

linear functions

A linear function is a type of function where the graph of the equation results in a straight line. These functions can be expressed in the form \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

• Slope \( (m) \): Reflects the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope means it falls.
• Y-intercept \( (b) \): The point where the line crosses the y-axis. Modifying \( b \) shifts the line up or down.

Linear functions are significant because they model relationships with constant rates of change, making them incredibly practical in real-life scenarios like predicting finances or measuring speed. Remember, linear functions that are vertical lines do not qualify as functions, as they fail the vertical line test.