(x%5E2%2B2)%2F(x%5E2%2B3)(x%5E2%2B4).jpeg "(x^2+1)(x^2+2)/(x^2+3)(x^2+4)")
Analyzing the Rational Function: (x^2+1)(x^2+2) / (x^2+3)(x^2+4)
This article will explore the properties of the rational function:
(x^2+1)(x^2+2) / (x^2+3)(x^2+4)
We will investigate its domain, potential asymptotes, and overall behavior.
Domain:
The domain of a rational function is restricted by the values of x that make the denominator zero. In this case, the denominator becomes zero when:
- x^2 + 3 = 0 => x^2 = -3 => This equation has no real solutions.
- x^2 + 4 = 0 => x^2 = -4 => This equation also has no real solutions.
Therefore, the denominator is never zero for real values of x. This means the domain of the function is all real numbers.
Asymptotes:
Since the degree of the numerator and denominator are both 4, we can determine the horizontal asymptote by comparing the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is also 1. Therefore, the horizontal asymptote is:
y = 1/1 = 1
The function does not have any vertical asymptotes as the denominator is never zero for real values of x.
Behavior:
Let's analyze the behavior of the function as x approaches positive and negative infinity:
- x approaching positive infinity: As x gets very large, the function will approach the horizontal asymptote y = 1 from above.
- x approaching negative infinity: As x gets very small, the function will approach the horizontal asymptote y = 1 from below.
Conclusion:
The rational function (x^2+1)(x^2+2) / (x^2+3)(x^2+4) is defined for all real numbers. It has a horizontal asymptote at y = 1 and no vertical asymptotes. Its behavior approaches the horizontal asymptote from above for large positive values of x and from below for large negative values of x.