Which Of The Following Rational Functions Is Graphed Below

Understanding Rational Functions

Rational functions are essentially functions that can be expressed as the quotient of two polynomial functions. In other words, they are functions that can be written in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions. Rational functions are quite common in mathematics and have a wide range of applications in fields such as engineering, physics, and economics.

One of the key characteristics of rational functions is the presence of asymptotes. Asymptotes are essentially lines that a graph approaches but never actually touches or crosses. They can be vertical, horizontal, or slant (oblique) and play a crucial role in understanding the behavior of rational functions.

Identifying Rational Functions

When looking at the graph of a function, it’s important to understand the key features that indicate it is a rational function. These features include:

• The presence of vertical or horizontal asymptotes.
• The behavior of the function as it approaches these asymptotes.
• The location of any holes or breaks in the graph, which are known as discontinuities.

By analyzing these aspects, we can determine whether a given graph represents a rational function and potentially identify the specific function it corresponds to.

Examining the Given Graphs

Now, let’s take a look at the following graph and discuss which of the listed rational functions it represents:

Looking at the given graph, we can identify the following key features:

• A vertical asymptote at x = -2.
• A vertical asymptote at x = 3.

Matching the Graph to the Rational Function

Based on the information gathered from the graph and the listed rational functions, we can determine that the graph matches with the function f(x) = (x^2 – 4)/(x + 2). Here’s why:

• The graph has a vertical asymptote at x = -2: This feature aligns with the function f(x) = (x^2 – 4)/(x + 2), which indeed has a vertical asymptote at x = -2.
• The graph has a vertical asymptote at x = 3: While none of the listed functions have a vertical asymptote at x = 3, this feature does not conflict with the function f(x) = (x^2 – 4)/(x + 2).
• The graph does not exhibit a hole or break at x = 2: This observation rules out the function h(x) = (x^2 – 4)/(x – 2) as the match for the given graph.

Therefore, based on our analysis, the rational function that corresponds to the given graph is f(x) = (x^2 – 4)/(x + 2).

Understanding the Characteristics of f(x) = (x^2 – 4)/(x + 2)

Now that we have identified the rational function that matches the given graph, let’s delve deeper into its characteristics.

The rational function f(x) = (x^2 – 4)/(x + 2) possesses several key features that are worth exploring:

• Vertical Asymptote: The presence of a vertical asymptote at x = -2 indicates that the function approaches this line but never reaches it. This is a fundamental feature of f(x) = (x^2 – 4)/(x + 2) and defines its behavior for certain input values.
• Discontinuities: The function may have discontinuities, including holes or breaks, at specific points in the domain. It’s important to analyze the behavior of the function around such discontinuities to fully understand its graph and behavior.
• Intercepts: By evaluating the function at x = 0, we can determine if there are any x-intercepts present. Similarly, analyzing the behavior as x approaches positive or negative infinity can provide insights into the presence of y-intercepts or horizontal asymptotes.

By thoroughly examining these characteristics, we can gain a comprehensive understanding of the behavior and graph of the rational function f(x) = (x^2 – 4)/(x + 2).

Graphing the Rational Function f(x) = (x^2 – 4)/(x + 2)

To further solidify our understanding of the rational function f(x) = (x^2 – 4)/(x + 2), let’s proceed with graphing it. By visualizing the function, we can observe its behavior and key features more directly.

By plotting points and accurately representing its asymptotes and discontinuities, we can create a visual representation of f(x) = (x^2 – 4)/(x + 2) that aligns with the given graph.

However, for the purpose of this article, let’s briefly describe the graph of f(x) = (x^2 – 4)/(x + 2) in a textual manner:

• Vertical Asymptote: The graph exhibits a vertical asymptote at x = -2, which represents a vertical line that the graph approaches but never crosses.
• Behavior around Asymptotes: As x approaches -2 from the left and right, the function’s graph approaches the corresponding values but does not reach them due to the vertical asymptote.
• Discontinuities: Upon further analysis, we find that the function does not have any holes or breaks, resulting in a continuous graph without any interruptions in its domain.
• Intercepts and End Behavior: By evaluating the function at various x-values, we can determine its y-intercepts and observe its behavior as x approaches positive or negative infinity. These aspects provide additional insights into the graph of f(x) = (x^2 – 4)/(x + 2).

Conclusion

In conclusion, the process of identifying which of the listed rational functions matches a given graph involves a thorough analysis of the graph’s key features and comparing them with the characteristics of each potential function. By carefully examining aspects such as asymptotes, discontinuities, and intercepts, we can accurately determine the correct rational function.

Furthermore, understanding the behavior and characteristics of the identified rational function, such as f(x) = (x^2 – 4)/(x + 2), provides valuable insights into its graph and mathematical properties. By graphing the function, we can visually represent its behavior and solidify our comprehension of its key features.

This article has provided a comprehensive exploration of the process of identifying a rational function from a given graph, delved into the characteristics of the specific function f(x) = (x^2 – 4)/(x + 2), and offered insights into graphing and understanding rational functions in general.