Question

How to write an equation for a rational function with : Horizontal asymptote at \[y = 5\]

Answer 1

Hint: In order to determine the equation for rational function with horizontal asymptote \[y = 5\].Assume the rations equation as\[y = \dfrac{{p\left( x \right)}}{{q\left( x \right)}}\]. Determine the $ q\left( x \right) $ such that it should not have any solution as there is no vertical asymptote. For the horizontal asymptote. Determine the $ p\left( x \right) $ such that the ratio of highest degrees of both \[p\left( x \right)\]and \[q\left( x \right)\]is equal to $ 5 $ .

Complete step-by-step answer:
We are given horizontal asymptote for some rational function as \[y = 5\].
Let’s assume a rational function \[y = \dfrac{{p\left( x \right)}}{{q\left( x \right)}}\]where x is the variable.
Since according to the question, the rational function has no vertical asymptote. So for no vertical asymptote the denominator should not have any solution.
Thus, $ q\left( x \right) $ is a function which has no solution or is in the form of sum of squares.
Let $ q\left( x \right) = {x^2} + 4 $
Now to have horizontal asymptote \[y = 5\], we should have $ p\left( x \right) $ such that the ratio between the highest degree of $ p\left( x \right) $ and $ q\left( x \right) $ equal to $ 5 $ .
For example let $ p\left( x \right) = 5{x^2} $
Hence, we have the rational function as $ y = \dfrac{{5{x^2}}}{{{x^2} + 4}} $ with horizontal asymptote as $ y = 5 $
Therefore, an equation of rational function with horizontal asymptote is $ y = \dfrac{{5{x^2}}}{{{x^2} + 4}} $ .
So, the correct answer is “ $ y = \dfrac{{5{x^2}}}{{{x^2} + 4}} $ .”.

Note: Remember that for the rational equation with horizontal asymptote $ y = 5 $ , you can have many rational equations. In the solution we have considered just an example. Students can also form different equations satisfying the conditions and requirements for the equation mentioned in the solution.
Vertical asymptotes for rational functions are found by setting the denominator equivalent to 0. This additionally assists with finding the domain. The domain can NOT contain that number