Question

find the rational expressions f(x):

Xintercepts (1/2,0) (-2,0)

y-intercepts (0,-2/3)

Holes: none

Vertical Asymptote x=3

Horyzontal y=6

Answer 1

When a rational function f(x) = p(x)/q(x),where p(x) and q(x) are polynomials in a single variable x, the equations of the vertical asymptotes can be found by finding the roots of q(x).

The horizontal asymptotes are determined by looking at the degrees of the numerator(n) and denominator(m). If n < m, the X-axis,i.e. the line y = 0 is the horizontal asymptote. If n= m, then y = a_n /b_m i.e. the ratio of the leading coefficients in p(x) and q(x), is the horizontal asymptote. If n >m, there is no horizontal asymptote.

The x-intercepts are determined by substituting y = 0 and the y- intercepts are determined by substituting x = 0.

Here, the x- intercepts of the required rational function f(x) = p(x)/q(x) ( say) are (1/2,0) and (-2,0). This implies that (x-1/2) and (x+2) are factors of the numerator p(x).

The line x = 3 is given to be the only vertical asymptote, hence (x-3) is a factor of the denominator q(x). Also, since there are no holes in the graph of f(x), it means that p(x) and q(x) do not have any common factors.

Further, the line y = 6 is   given to be the only vertical asymptote, hence the degree of the numerator is same as the degree of the denominator. Also, the ratio of the leading coefficients in p(x) and q(x) is 6.

Therefore, the required rational function is y = f(x) = 6(x-1/2)(x+2)/(x-3)². On substituting x = 0, we get y = 6*(-1/2)*2/(-3)² = -6/9 = -2/3.

Thus, the rational function y = f(x) = 6(x-1/2)(x+2)/(x-3)² satisfies all the given conditions.