Chapter 11.7, Problem 5LC

To determine

Write an example of a rational function with a vertical asymptote at $x=-2$ and a horizontal asymptote at $y=4$ .

Answer to Problem 5LC

  $f\left(x\right)=\frac{8}{x+2}+4$

Explanation of Solution

Given information:

Vertical asymptote at $x=-2$

Horizontal asymptote at $y=4$

Concept used:

• The vertical asymptotes of a function occur at the points where the function is undefined. Vertical asymptote is the vertical line that the graph of rational function follows till infinity but never intersects.

The horizontal asymptote of a rational function is given by:

• If the degree of the polynomial in numerator is less than the degree of polynomial in denominator, then the horizontal asymptote is the horizontal axis or y=0.
• If the degree of the polynomial in numerator is equal to the degree of polynomial in denominator, then the horizontal asymptote is the ratio of the leading coefficients of numerator and denominator.
• If the degree of the polynomial in numerator is greater than the degree of polynomial in denominator, then there is no horizontal asymptote. In this case there occurs a slant asymptote.
• We know that the vertical asymptotes occur at the values of x for which the denominator is zero.

Calculation:

The Rational function with a vertical asymptote at $x=-2$ and a horizontal asymptote at $y=4$ is

  $f\left(x\right)=\frac{8}{x+2}+4$

Vertical asymptote $x+2=0$

  $x=-2$

Horizontal asymptote at $y=4$