%2F(x%5E2%2B4).jpeg "(x^2-4)/(x^2+4)")
Exploring the Rational Function: (x^2 - 4)/(x^2 + 4)
This article delves into the properties and characteristics of the rational function (x^2 - 4)/(x^2 + 4), exploring its key features, such as its domain, asymptotes, and graph.
Domain
The domain of a rational function is determined by the values of x that make the denominator non-zero. In this case, the denominator (x^2 + 4) is never equal to zero, as the square of any real number is always non-negative, and adding 4 makes it strictly positive.
Therefore, the domain of this function is all real numbers, represented as (-∞, ∞).
Asymptotes
Horizontal Asymptote
To find the horizontal asymptote, we examine the degrees of the numerator and denominator. Both the numerator and denominator have a degree of 2. Since the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients.
The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 1.
Vertical Asymptote
Vertical asymptotes occur where the denominator of the rational function equals zero. As we established earlier, the denominator (x^2 + 4) is never zero. Therefore, there are no vertical asymptotes for this function.
Graphing the Function
To visualize the function, we can consider a few key points and the behavior near the asymptotes:
- x = 0: The function evaluates to (-4/4) = -1. This is the y-intercept.
- x → ∞ and x → -∞: As x approaches positive or negative infinity, the function approaches the horizontal asymptote y = 1.
- Symmetry: The function is even because it is symmetrical about the y-axis. This means f(x) = f(-x).
Considering these points and the absence of vertical asymptotes, we can sketch a graph that shows the function approaching the horizontal asymptote from both sides, with a dip at the y-intercept.
Conclusion
The rational function (x^2 - 4)/(x^2 + 4) has a domain of all real numbers, a horizontal asymptote at y = 1, and no vertical asymptotes. Its graph exhibits symmetry about the y-axis and approaches the horizontal asymptote as x tends towards positive or negative infinity. This function provides a good example of a rational function with a horizontal asymptote and a relatively simple graph to analyze.