Write the equation of a possible rational function with the following characteristics. Explain your reasoning.
Vertical asymptotes
$a) \quad$ Vertical asymptotes at $\;x = \pm 3$, $x$-intercepts at $x = 5 \;\;and \; x = -1,$ and a horizontal
$\quad\quad$ asymptote of $y = \dfrac 12$
$b) \quad$ Vertical asymptotes at $x = \dfrac 14,\;$ $x$-intercept of $\,x = 0,\;$ and a discontinuous point at $\,\left(5,\dfrac 5{19}\right)$
$c) \quad$ $Y$-intercept at $-5, \,$ no $x$-intercepts, discontinuous points at $\,(-1, -5)\;and\; (3, -5)$
Best Answer
(a). Vertical asymptotes at $x=±3$ imply that the denominator could be $x^2-9$, $x$-intercepts at $x=5$ and $x=−1$ imply that the numerator could be $(x-5)(x+1)$, and a horizontal asymptote of $y=\frac{1}{2}$ implies that $\lim_{x\to\infty}f(x)=\frac{1}{2}$. Putting all the argument together, the function is $$ f(x)=\frac{(x-5)(x+1)}{2(x^2-9)}. $$ You can use the same reasoning for (b) and (c). Now I do not have time and I will come back tomorrow.