Question

Create a function whose graph has the given characteristics. (There are many correct answers.) Vertical asymptote: \(x=5\) Horizontal asymptote: \(y=0\)

Short answer

The function \(f(x) = \frac{1}{x-5}\) has a vertical asymptote at \(x=5\) and a horizontal asymptote at \(y=0\) as required.

Step-by-step solution

Formulate the Function

A simple way to formulate a function with these asymptotes is to make the function a fraction where \(x-5\) is in the denominator and a constant is in the numerator. This will create the vertical asymptote at \(x=5\). The function will approach 0 as \(x\) approaches positive or negative infinity due to the degree of the denominator being greater than the numerator. An example function with these properties can be \(f(x) = \frac{1}{x-5}\).

Check the Vertical Asymptote

Let's check that this function has a vertical asymptote at \(x=5\). As \(x\) approaches 5 from the left or right, the function will approach negative or positive infinity respectively. This confirms the presence of a vertical asymptote at \(x=5\).

Check the Horizontal Asymptote

It's also necessary to confirm that this function has a horizontal asymptote at \(y=0\). As \(x\) approaches either positive or negative infinity, the value of the function will approach 0. Thus, the function does have a horizontal asymptote at \(y=0\).