Create an equation for a A rational function not a polynomia

Create an equation for a. A rational function (not a polynomial) with no vertical asymptotes. b. A rational function with asymptotes y = 0, x = 2, x = -3 c. A rational function with asymptotes y = -1/3, x = 4, x = -6 and zeros at 2 and 7 d. A rational function with asymptotes y = 6/5, x = 6, x = -3 and zero at 1 only

Solution

a) a rational function with no vertical asymptotes

a rational function will have no vertical asymptotes depending upon whether the denominator has zeros or not

f(x) = (x+3) / (x^2 + 16)

in this function denominator has no zeros hence, this function has no vertical asymptotes

b) rational function with asymptotes y = 0 , x = 2 , x = -3

so in this case degree of denominator shouls be greater than that of numerator as horizontal asymptote is y = 0

therefore, function is

f(x) = ( x-3 ) / (x^2 + x - 6 )

c) asymptotes y = -1/3 , x = 4 ,x = -6 and zeros at 2 and 7

function becomes

the zeros come in the numerator and vertical asymptotes in denominator

f(x) = -1/3 ( x-2)(x-7) / (x-4)(x+6)

d) asymptotes y = 6/5 , x = 6 , x = -3 and zero at 1

the zeros come in the numerator and vertical asymptotes in denominator

f(x) = 6/5 (x-1)^2 / (x-6)(x+3)