Problem 31 - Algebra 2

Question

Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function and state whether it is discrete or continuous. $$ y=3 x-4 $$

Step-by-Step Solution

Answer

The line is a function, with domain \(x \in \mathbb{R}\), range \(y \in \mathbb{R}\), and it is continuous.

Step 1: Understand the Equation

The given equation is a linear equation in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = 3\) and \(b = -4\).

Step 2: Graph the Equation

To graph \(y = 3x - 4\), start by plotting the y-intercept \((0, -4)\) on the coordinate plane. Then use the slope \(3\), which means 'rise over run' or '3 over 1', to find another point. From \((0, -4)\), move up 3 units and 1 unit to the right to locate another point, \((1, -1)\). Draw a straight line that passes through these points.

Step 3: Determine the Domain

For any linear equation like \(y = 3x - 4\), the domain is all real numbers. This means \(x\) can take any real value. We write: Domain: \(x \in \mathbb{R}\).

Step 4: Determine the Range

Similarly, for a linear equation, the range is also all real numbers because \(y\) can take any value along the line. We write: Range: \(y \in \mathbb{R}\).

Step 5: Check if it is a Function

A relation is a function if every \(x\)-value has exactly one \(y\)-value. In the equation \(y = 3x - 4\), for every \(x\), there is exactly one corresponding \(y\), thus it is a function.

Step 6: Determine if it is Discrete or Continuous

Since the equation represents a line without breaks, it is continuous. We plot a continuous line through the points.

Key Concepts

Domain and Range

When graphing equations, two critical concepts are domain and range. They help us understand which values the variables can take:

Domain refers to all possible inputs (or x-values) the equation can have. For linear equations like \(y = 3x - 4\), the domain is all real numbers because you can substitute any real number for \(x\). In mathematical terms, this is denoted as \(x \in \mathbb{R}\).

Range, on the other hand, involves all possible outputs (or y-values) the equation can produce. For the same linear equation, since the line extends indefinitely in both directions, the range also includes all real numbers, written as \(y \in \mathbb{R}\). In simpler words, for every \(x\), there can be a corresponding real number \(y\).

These concepts are essential when defining how broad or narrow the values can be for an equation. Understanding domain and range helps in grasping the extent and limitations of a relation or function.

Functions

Functions are a fundamental concept in mathematics. They express a unique relationship where each x-value has exactly one y-value. A simple way to check if a relationship is a function is to use the vertical line test. If a vertical line drawn through the graph of the relation touches the curve at more than one point, it is not a function.

In the given equation \(y = 3x - 4\), we notice that for every \(x\), there is only one \(y\) produced. This characteristic ensures that the relationship qualifies as a function.

Functions are crucial because they provide a clear and consistent connection between variables. They help easily predict the behavior of variables over their domains. This predictability is what makes functions so valuable across different areas in mathematics and science, making them a backbone for graphing equations.

Discrete vs Continuous

Understanding the nature of graphs is key to solving and interpreting equations. Graphs can be either discrete or continuous based on their properties.

A discrete graph consists of isolated points. Each value is separate, and there are gaps between points. Discrete graphs occur when the data or values cannot be finely divided, such as plotting whole numbers or counts.

On the other hand, a continuous graph is a smooth, unbroken line or curve that represents all values within a given interval. In the equation \(y = 3x - 4\), the graph is continuous because every value of \(x\) leads to a corresponding value of \(y\) without any interruptions.

This continuity means that you can select any real number for \(x\), leading to a complete, connected line across the plane. Recognizing whether a graph is discrete or continuous helps in understanding the scope and applications of different equations.