Question

How do you write a rational function that has the following properties: a zero at x= 4, a hole at x= 7, a vertical asymptote at x= -3, a horizontal asymptote at y= $\dfrac{2}{5}$?

Answer 1

To solve this question, we will use some of the properties of a rational function. If a function has a zero at $x=a$, then it has $\left( x-a \right)$ as a factor in the numerator. If a function has a hole at $x=a$, then it has $\left( x-a \right)$ as a factor in numerator and denominator. If a function has a vertical asymptote at $x=a$, then it has $\left( x-a \right)$ as a factor in its denominator. If a function has a vertical asymptote at $y=a$, then it means that the highest degrees in numerator and denominator are equal and their coefficients are in the ratio of $a:1$.

Complete step by step answer:
We are asked to write a function that has the given properties. We will look at each of the properties one by one and make the function accordingly.
The properties say that the function has a zero at $x=4$, which means that the function has $\left( x-4 \right)$ as a factor in the numerator. A hole at $x=7$ means that it has $\left( x-7 \right)$ a factor in the numerator as well as denominator. A vertical asymptote at $x=-3$ means $\left( x+3 \right)$ a factor in the denominator only. A horizontal asymptote at $y=\dfrac{2}{5}$ means highest degrees in both numerator and denominator are equal and their coefficients are in ratio of $2:5$.
Using all these properties we get the desired function as $\dfrac{2\left( x-4 \right)\left( x-7 \right)}{5\left( x-7 \right)\left( x+3 \right)}$. Simplifying the function, we get,
$\dfrac{2{{x}^{2}}-22x+56}{5{{x}^{2}}-20x-105}$ is the desired function.

Note:
These questions can be solved by remembering the properties of a function. The properties include zeroes, vertical asymptote, horizontal asymptote, hole etc. The solution can be verified by plotting the graph of the function, and confirming whether it has the given properties or not.