A rational function is the function that can be written as the ratio of two polynomials. Rational function can be divided into proper and improper function depending on the highest degree at numerator, n and denominator, m. Given R(x) in the form of R(x) = p1xn + q1xn-1 + r1xn-2+ s1 / p2xm + q2xm-1 + r2xm-2 + s2 where p, q, r and s are nonzero and positive integers. a. Develop TWO (2) R(x) functions that satisfies the given conditions below: i. Function 1: when n < m ii. Function 2: when n > m , where m = n − 1. and given 2 ≤ n ≤ 4 and 2 ≤ m ≤ 4 . then, compare and discuss the behavior of the functions based on the following details: • the type of rational functions (i.e: proper or improper) • the asymptotes (i.e: vertical asymptote, horizontal asymptote or oblique asymptote) • the intercepts (i.e: x-intercept and y-intercept) • the graphs of the functions.
A rational function is the function that can be written as the ratio of two polynomials. Rational function can be divided into proper and improper function depending on the highest degree at numerator, n and denominator, m. Given R(x) in the form of
R(x) = p1xn + q1xn-1 + r1xn-2+ s1 / p2xm + q2xm-1 + r2xm-2 + s2
where p, q, r and s are nonzero and positive integers.
a. Develop TWO (2) R(x) functions that satisfies the given conditions below:
i. Function 1: when n < m
ii. Function 2: when n > m , where m = n − 1. and given 2 ≤ n ≤ 4 and 2 ≤ m ≤ 4 .
then, compare and discuss the behavior of the functions based on the following details:
• the type of rational functions (i.e: proper or improper)
• the asymptotes (i.e: vertical asymptote, horizontal asymptote or oblique asymptote)
• the intercepts (i.e: x-intercept and y-intercept)
• the graphs of the functions.
Show all the appropriate steps needed for the required details above. Hence, analyze the relationship between the vertical asymptote x = cn, n = 1,2,3,4 and the behavior of the functions, R(x) as x is approaching cn.
limx→cn+ ->R(x) , limx→cn-−> R(x).
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