Analysis Of The Rational Function (2x^2 + 2) / (4 - X^2)

In this comprehensive exploration, we delve into the intricacies of the rational function (2x^2 + 2) / (4 - x^2). Rational functions, formed by the ratio of two polynomials, play a vital role in various fields of mathematics, science, and engineering. Understanding their behavior, characteristics, and graphical representation is crucial for solving real-world problems and developing a deeper understanding of mathematical concepts. This article will provide a detailed analysis of the given rational function, covering essential aspects such as domain, intercepts, asymptotes, symmetry, and graphical representation. By examining these properties, we aim to gain a thorough understanding of the function's behavior and its applications.

The domain of a rational function is the set of all real numbers except for those values that make the denominator equal to zero. These values are excluded because division by zero is undefined. For the function (2x^2 + 2) / (4 - x^2), the denominator is 4 - x^2. To find the values that make the denominator zero, we set it equal to zero and solve for x:

4 - x^2 = 0 x^2 = 4 x = ±2

Therefore, the domain of the function is all real numbers except x = 2 and x = -2. In interval notation, the domain can be expressed as (-∞, -2) ∪ (-2, 2) ∪ (2, ∞). Understanding the domain is crucial because it identifies the values for which the function is defined and helps us avoid undefined points in our analysis. The domain also informs us about potential vertical asymptotes, which occur at the values excluded from the domain.

Intercepts are points where the graph of a function intersects the coordinate axes. There are two types of intercepts: x-intercepts and y-intercepts. X-intercepts are the points where the graph intersects the x-axis, and they occur when the function's value (y) is zero. Y-intercepts are the points where the graph intersects the y-axis, and they occur when x is zero. To find the x-intercepts, we set the function equal to zero and solve for x:

(2x^2 + 2) / (4 - x^2) = 0

For a fraction to be zero, the numerator must be zero. So, we set the numerator equal to zero:

2x^2 + 2 = 0 2x^2 = -2 x^2 = -1

Since the square of a real number cannot be negative, there are no real solutions for x. This means that the function has no x-intercepts. To find the y-intercept, we set x = 0 in the function:

y = (2(0)^2 + 2) / (4 - (0)^2) y = 2 / 4 y = 1/2

Thus, the y-intercept is (0, 1/2). The absence of x-intercepts and the presence of a y-intercept at (0, 1/2) provide valuable information about the function's graph and behavior. The graph will cross the y-axis at 1/2 but will not intersect the x-axis.

Asymptotes are lines that the graph of a function approaches but never touches. They are crucial in understanding the end behavior and shape of rational functions. There are three types of asymptotes: vertical, horizontal, and oblique (or slant). Vertical asymptotes occur at the values excluded from the domain, where the denominator of the rational function is zero. We already found that the domain excludes x = 2 and x = -2. Therefore, the function has vertical asymptotes at x = 2 and x = -2. Horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity. To find the horizontal asymptote, we compare the degrees of the numerator and denominator. In this case, the numerator is 2x^2 + 2 (degree 2), and the denominator is 4 - x^2 (degree 2). Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients:

y = 2 / -1 y = -2

Thus, the function has a horizontal asymptote at y = -2. Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the degrees are equal, so there is no oblique asymptote. Identifying the asymptotes is fundamental to sketching the graph of the function. Vertical asymptotes define the boundaries where the function approaches infinity, while horizontal asymptotes indicate the long-term behavior of the function.

Symmetry refers to the property of a function's graph being identical on both sides of a line or point. There are two types of symmetry to consider: even and odd. A function is even if f(-x) = f(x) for all x in the domain. The graph of an even function is symmetric with respect to the y-axis. A function is odd if f(-x) = -f(x) for all x in the domain. The graph of an odd function is symmetric with respect to the origin. To determine if the function (2x^2 + 2) / (4 - x^2) is even or odd, we evaluate f(-x):

f(-x) = (2(-x)^2 + 2) / (4 - (-x)^2) f(-x) = (2x^2 + 2) / (4 - x^2)

Since f(-x) = f(x), the function is even. This means the graph of the function is symmetric with respect to the y-axis. Recognizing the symmetry of the function simplifies the process of graphing because we only need to analyze one side of the y-axis.

To graph the rational function (2x^2 + 2) / (4 - x^2), we use the information we have gathered: the domain, intercepts, asymptotes, and symmetry. We know the domain is (-∞, -2) ∪ (-2, 2) ∪ (2, ∞), which gives us vertical asymptotes at x = -2 and x = 2. The y-intercept is (0, 1/2), and there are no x-intercepts. The horizontal asymptote is y = -2, and the function is symmetric with respect to the y-axis. Using this information, we can sketch the graph. The graph will approach the vertical asymptotes at x = -2 and x = 2. As x approaches infinity or negative infinity, the graph will approach the horizontal asymptote y = -2. The y-intercept provides a point on the graph, and the symmetry helps us mirror the graph on both sides of the y-axis. To get a more accurate graph, we can plot a few additional points. For example, we can evaluate the function at x = 1 and x = -1:

f(1) = (2(1)^2 + 2) / (4 - (1)^2) = 4 / 3 f(-1) = (2(-1)^2 + 2) / (4 - (-1)^2) = 4 / 3

So, the points (1, 4/3) and (-1, 4/3) are on the graph. The graph will consist of three separate sections, divided by the vertical asymptotes. In the interval (-∞, -2), the graph will be below the horizontal asymptote and approach the vertical asymptote x = -2 from the left. In the interval (-2, 2), the graph will pass through the y-intercept (0, 1/2) and be above the x-axis, approaching the vertical asymptotes x = -2 and x = 2. In the interval (2, ∞), the graph will be below the horizontal asymptote and approach the vertical asymptote x = 2 from the right. The graph visually represents the function's behavior and confirms our analysis of its properties.

In conclusion, analyzing the rational function (2x^2 + 2) / (4 - x^2) involves a detailed examination of its domain, intercepts, asymptotes, symmetry, and graphical representation. The domain excludes x = 2 and x = -2, leading to vertical asymptotes at these values. The function has a y-intercept at (0, 1/2) but no x-intercepts. The horizontal asymptote is y = -2, and the function is even, exhibiting symmetry with respect to the y-axis. The graph consists of three separate sections, approaching the asymptotes and reflecting the symmetry. Understanding these properties is essential for comprehending the behavior of rational functions and their applications in various mathematical and scientific contexts. By systematically analyzing the function, we have gained a comprehensive understanding of its characteristics and graphical representation. This thorough analysis provides valuable insights into the nature of rational functions and their role in mathematical problem-solving.