Question

Does any of the foltowing, drawn in a rectangular coordinate plane, represent a function? (a) A circle (b) A triangle (c) A rectangle (d) A downward-sloping straight line

Short answer

Only a downward-sloping straight line represents a function.

Step-by-step solution

Understand the Definition of a Function

A function is a relation in which each input (from the domain) is related to exactly one output (from the range). In the coordinate plane, this means that for every value of \(x\), there must be exactly one corresponding value of \(y\). If a vertical line intersects the graph at more than one point, then it is not a function. This is known as the vertical line test.

Analyze a Circle

The graph of a circle doesn't pass the vertical line test. For some values of \(x\), a vertical line will intersect the circle at two points, therefore making a circle not a function.

Analyze a Triangle

Similar to a circle, for certain values of \(x\), a vertical line can intersect a triangle at more than one point. Therefore, a triangle is generally not a function, unless it is tilted such that it passes the vertical line test.

Analyze a Rectangle

For a rectangle, vertical lines will intersect the figure twice, except the lines that are extending only on the vertical sides. Since multiple \(y\) values correspond to single \(x\) values (where the line intersects), a rectangle too is not a function.

Analyze a Downward-Sloping Straight Line

A straight line, regardless of its slope, will pass the vertical line test because for each \(x\) there is only one corresponding \(y\). Therefore, a downward-sloping straight line represents a function.

Key concepts

Vertical Line Test

The vertical line test is a handy method to visually determine whether a curve on a coordinate plane represents a function. This test is straightforward and can save time when trying to identify functions. Here's how it works: imagine drawing vertical lines on the graph. As you slide these vertical lines across the coordinate plane, observe how many points the line intersects on the curve.

• If each vertical line touches the curve at exactly one point, then the graph is a function.
• If any vertical line crosses the curve at more than one point, then the graph is not a function.

This test is rooted in the definition of a function, where each input must map to a single output. It's essential because it translates the concept of functions from algebra into a visual form that is easier to understand.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional flat surface formed by two perpendicular number lines called axes. The horizontal axis is known as the x-axis, while the vertical axis is the y-axis. Where they intersect is called the origin, denoted by the point (0,0). The coordinate plane is crucial in graphing equations and visualizing geometric shapes. Each point on the plane is represented by a pair of numbers (x, y), which are the coordinates. These coordinates tell you exactly where a point is located relative to the origin.

• The x-coordinate shows the position left or right of the origin.
• The y-coordinate shows the position above or below the origin.

Using the coordinate plane helps in visualizing mathematical concepts and solving problems by providing a graphical representation of functions and relations.

Geometry

Geometry deals with the properties, measurements, and relationships of points, lines, surfaces, and solids. In the context of determining functions, we frequently deal with geometric shapes such as circles, triangles, and rectangles. Understanding the geometric perspective is crucial when applying the vertical line test:

• In a circle or a rectangle, a vertical line can intersect the shape at multiple points, implying they are not functions.
• Triangles may sometimes meet the criteria of a function if their orientation aligns such that no vertical line crosses them more than once.

In geometry, visual methods can greatly aid in understanding abstract mathematical concepts, bridging the gap between theoretical math and practical visualization.

Relation and Function

A relation in mathematics is a set of ordered pairs, while a function is a specific type of relation. For a relation to be a function, each input or x-value must correspond to exactly one output or y-value. When determining if a relation is a function using its graph, we apply the vertical line test as a quick check. This method helps confirm if for every x, there is a unique y. Consider several geometric figures:

• A downward-sloping straight line is a function because each x maps to only one y.
• Circles, rectangles, and untitled triangles typically do not qualify as functions since they fail the vertical line test.

Understanding the distinction between general relations and functions emphasizes the unique relationship functions have, and why precise identification of these in graphs is crucial in mathematics.