Problem 54 - Algebra 2

Question

Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function. $$ (3,5),(4,6),(5,-4) $$

Step-by-Step Solution

Answer

Domain: \( \{3, 4, 5\} \); Range: \( \{5, 6, -4\} \); It is a function.

Step 1: Plot the Points

Start by graphing the points given in the relation on a coordinate plane. The points are (3, 5), (4, 6), and (5, -4). Place a dot at each of these coordinates.

Step 2: Determine the Domain

The domain of a relation is the set of all possible x-values. From the points given, the x-values are 3, 4, and 5. Therefore, the domain is \( \{3, 4, 5\} \).

Step 3: Determine the Range

The range of a relation is the set of all possible y-values. From the points given, the y-values are 5, 6, and -4. Thus, the range is \( \{5, 6, -4\} \).

Step 4: Check if the Relation is a Function

To determine if a relation is a function, check that each x-value is paired with exactly one y-value. In this relation, each x-value 3, 4, and 5 is paired with only one y-value (5, 6, and -4, respectively). Thus, this relation is a function.

Key Concepts

Domain and Range

In any mathematical relation or equation, the **domain** refers to the set of all possible input values, often represented by the variable \( x \). Essentially, it's all the numbers you can plug into a function which will, in return, give a valid output. In our example exercise, the domain consists of the x-values from the set of ordered pairs, which are \( 3 \), \( 4 \), and \( 5 \).
On the other hand, the **range** represents all possible output values, commonly shown as the variable \( y \). For our example, the range includes the y-values \( 5 \), \( 6 \), and \( -4 \).
Understanding domain and range is crucial because they tell us the scope of a function and its limitations or applicability. They determine which numbers can work as inputs and what outputs they will produce.

Graphing Points

Graphing points on a coordinate plane is a foundational skill in understanding and solving mathematical relations. When you face an equation or set of points like the exercise example \((3,5), (4,6), (5,-4)\), you start by identifying each point's coordinates.
The first number in each pair is the x-coordinate, and the second one is the y-coordinate. You can plot points by:

• Moving horizontally to the x-coordinate on the graph.
• Then, moving vertically to reach the y-coordinate.

Once plotted, you connect the points if the context requires it or simply ensure each is marked clearly on the plane. This visual representation helps in analyzing the characteristics and behavior of the relation or function.

Coordinate Plane

The coordinate plane is a two-dimensional surface used for graphing points, lines, and curves. It consists of two perpendicular axes:

• The horizontal axis is known as the x-axis.
• The vertical axis is called the y-axis.

The point where these axes intersect is called the origin, typically denoted as \((0,0)\).
Each point on the plane can be described with an ordered pair \((x, y)\). The x-coordinate tells you the position along the horizontal axis while the y-coordinate shows the placement on the vertical axis.
Understanding the coordinate plane is essential as it allows you to visualize mathematical concepts graphically, making it easier to analyze and solve equations.

Determining Functions

Identifying whether a set of points or a relation is a function is vital. A **function** is a type of relation where each input \( x \) corresponds to exactly one output \( y \). This means none of the x-values can be associated with more than one y-value.
To determine if a relation is a function, simply check each x-value to ensure it has only one y-value. In the example with points \((3,5), (4,6), (5,-4)\):

• \( x = 3 \) maps to \( y = 5 \)
• \( x = 4 \) maps to \( y = 6 \)
• \( x = 5 \) maps to \( y = -4 \)

Since all x-values have unique y-values, the given relation is indeed a function. Understanding functions is fundamental in math as they describe consistent and predictable relationships between variables.