Question

Give an example of a rational function that has a horizontal asymptote at y = 0 and a vertical asymptote atx = 2 and x = 1.

Answer 1

Step 1

Horizontal Asymptotes of Rational Functions

The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.

If N is the degree of the numerator and D is the degree of the denominator, and…

N < D, then the horizontal asymptote is y = 0.

N = D, then the horizontal asymptote is y = ratio of leading coefficients.

N > D, then there is no horizontal asymptote.

Step 2

Identify Vertical Asymptotes of a Rational Function

Factor the numerator and denominator.

Simplify by canceling common factors in the numerator and the denominator.

Set the simplified denominator equal to zero and solve for x.

Step 3

x = 2 and x = 1

x - 2 and x - 1

The denominator expression will be (x-2)(x-1)

Step 4

`\begin{gathered} The\text{ rational fraction is} \\ \\ y=(1)/((x-2)(x-1)) \end{gathered}`

Final answer

`y\text{ = }(1)/((x-2)(x-1))`