Chapter 2.6, Problem 80E

To determine

To write: a rational function f whose graph has no vertical asymptotes and has horizontal asymptote at $y=2$ .

Answer to Problem 80E

  $f\left(x\right)=\frac{2x^2+5}{x^2+1}$

Explanation of Solution

Concept Used:

Vertical Asymptote:

“The line $x=a$ is a vertical asymptote of the graph of a rational function $f\left(x\right)=\frac{N\left(x\right)}{D\left(x\right)}$ , only if ‘ a’ is a zero of $D\left(x\right)$ .”

Horizontal Asymptote:

“For the given rational function $f\left(x\right)=\frac{N\left(x\right)}{D\left(x\right)}$ where degree of $N\left(x\right)$ is m and degree of $D\left(x\right)$ is n .

1. The line $y=0$ is a horizontal asymptote if $m<n$ .

2. The line $y=\frac{a_m}{b_n}$ (the ratio of leading coefficients) is a horizontal asymptote if $m=n$ .

3. There is no horizontal asymptote if $m>n$ .”

Since, the required function has no vertical asymptote, it means that the denominator of this function have no real roots.

So, let $D\left(x\right)=x^2+1$

Also, the function has horizontal asymptote at $y=2$ , this implies the degree of the numerator of the function must be equal to the degree of its denominator. And the ratio of the leading coefficients must be 2.

Now the leading coefficient of $D\left(x\right)$ is 1. So, it implies that the leading coefficient of $N\left(x\right)$ must be 2. Also, the degree of $N\left(x\right)$ must be 2.

So, let $N\left(x\right)=2x^2+5$

Then, the required rational function is $f\left(x\right)=\frac{2x^2+5}{x^2+1}$ .