Graphing And Analyzing F(x) = -8 + 1/x

Nov 19, 2025 by Admin 39 views

Hey guys! Today, we're going to dive into graphing and analyzing the rational function F(x) = -8 + 1/x. We'll break it down step by step, using transformations to sketch the graph, determining the domain and range, and identifying any asymptotes. Let's get started!

(a) Graphing the Rational Function Using Transformations

To graph the rational function F(x) = -8 + 1/x using transformations, we'll start with the basic reciprocal function, y = 1/x, and apply a series of transformations. Understanding these transformations is key to accurately graphing the function. This involves recognizing how each part of the equation affects the graph's position and orientation on the coordinate plane. Mastering transformations will not only help in graphing this specific function but also in understanding and graphing a wide range of other functions.

1. Start with the Basic Reciprocal Function: y = 1/x

The basic reciprocal function, y = 1/x, has a simple yet distinctive shape. It consists of two branches, one in the first quadrant and one in the third quadrant. As x approaches zero, y approaches infinity, creating a vertical asymptote at x = 0. Similarly, as x approaches infinity, y approaches zero, creating a horizontal asymptote at y = 0. This foundational graph serves as the starting point for our transformations. Understanding its key features, such as the asymptotes and the behavior of the branches, is crucial before applying any transformations. The graph is symmetric with respect to the origin.

2. Vertical Shift: F(x) = -8 + 1/x

The given function F(x) = -8 + 1/x can be seen as a vertical shift of the basic reciprocal function y = 1/x. Specifically, the '+ -8' part of the equation indicates that the entire graph of y = 1/x is shifted downward by 8 units. This means that every point on the original graph is moved down 8 units on the coordinate plane. The horizontal asymptote, which was originally at y = 0, is also shifted down 8 units to y = -8. This vertical shift fundamentally changes the position of the graph without altering its shape or orientation. Consequently, the graph now exists relative to this new horizontal asymptote, influencing our understanding of the function's behavior as x approaches infinity.

Graphing Steps

Sketch y = 1/x: Draw the basic reciprocal function with asymptotes at x = 0 and y = 0.
Shift Down 8 Units: Move the entire graph of y = 1/x down 8 units. The new horizontal asymptote is y = -8.

By following these steps, you'll accurately graph the rational function F(x) = -8 + 1/x. Understanding the transformations involved allows for a clear visualization of the function's behavior and characteristics.

(b) Domain and Range

Alright, let's figure out the domain and range of F(x) = -8 + 1/x using the graph we just created. The domain and range are essential properties that describe the set of all possible input and output values for the function. Understanding these properties helps in comprehending the function's behavior and limitations. Accurately determining the domain and range is crucial for various mathematical applications, including solving equations, analyzing function behavior, and modeling real-world phenomena.

Domain

The domain of a function is the set of all possible x-values (inputs) for which the function is defined. For the rational function F(x) = -8 + 1/x, we need to identify any values of x that would make the function undefined. Rational functions are undefined when the denominator is equal to zero. In this case, the denominator is x, so the function is undefined when x = 0. Therefore, the domain of F(x) is all real numbers except x = 0. This can be expressed in interval notation as (-∞, 0) ∪ (0, ∞). Identifying such restrictions is critical in many mathematical contexts, particularly when dealing with division or other operations that could potentially lead to undefined results.

Range

The range of a function is the set of all possible y-values (outputs) that the function can produce. From the graph of F(x) = -8 + 1/x, we can see that the function can take on any y-value except for y = -8, which is the horizontal asymptote. As x approaches infinity, F(x) approaches -8, but it never actually reaches -8. Therefore, the range of F(x) is all real numbers except y = -8. In interval notation, this is expressed as (-∞, -8) ∪ (-8, ∞). Determining the range often involves analyzing the function's end behavior and any restrictions imposed by asymptotes or other features of the function.

(c) Asymptotes

Now, let's identify the asymptotes of the function F(x) = -8 + 1/x. Asymptotes are lines that the graph of a function approaches but never touches. They provide important information about the function's behavior as x approaches certain values or infinity. There are three types of asymptotes: vertical, horizontal, and oblique. In the case of rational functions, asymptotes often occur where the function is undefined or as x approaches very large or very small values.

Vertical Asymptotes

Vertical asymptotes occur where the function is undefined. For F(x) = -8 + 1/x, the function is undefined when the denominator x is equal to zero. Therefore, there is a vertical asymptote at x = 0. The graph approaches this line but never intersects it. Understanding the vertical asymptotes helps in identifying where the function has discontinuities and provides insights into its behavior near these points.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity. For F(x) = -8 + 1/x, as x becomes very large (either positive or negative), the term 1/x approaches zero. Therefore, F(x) approaches -8. This means there is a horizontal asymptote at y = -8. The graph approaches this line as x goes to ±∞. The presence of a horizontal asymptote indicates the long-term behavior of the function, showing what value the function tends toward as the input becomes extremely large or small.

Oblique Asymptotes

Oblique asymptotes (also called slant asymptotes) occur when the degree of the numerator is exactly one greater than the degree of the denominator. In the case of F(x) = -8 + 1/x, the function can be rewritten as F(x) = (-8x + 1) / x. The degrees of the numerator and denominator are both 1. Since the degree of the numerator is not exactly one greater than the degree of the denominator (it's equal), there is no oblique asymptote. Oblique asymptotes are important for understanding the behavior of certain rational functions that do not have horizontal asymptotes.

In summary:

Vertical Asymptote: x = 0
Horizontal Asymptote: y = -8
Oblique Asymptote: None

And there you have it! We've graphed the function, found its domain and range, and identified all its asymptotes. Hope this helped you guys out!