Question

Write a rational function f(x) that has the specific characteristics:

1. Vertical Asymptote at x = 2.
2. Horizontal Asymptote at y = 1/2.
3. Zero at x = 1.

Answer 1

A rational function that satisfies the given conditions is f(x) = (x - 1)/(2(x - 2)), which includes a zero at x = 1, a vertical asymptote at x = 2, and a horizontal asymptote at y = 1/2.

Step-by-step explanation:

To construct a rational function, f(x), with a vertical asymptote at x = 2, a horizontal asymptote at y = 1/2, and a zero at x = 1, one would need to set a denominator that becomes zero when x = 2, and a numerator that becomes zero when x = 1. Moreover, to ensure the horizontal asymptote at y = 1/2, the degrees of the numerator and denominator must be the same with appropriate leading coefficients.

Based on these requirements, one possibility for f(x) could be:

\( f(x) = \frac{(x - 1)}{2(x - 2)} \)

This function has a zero at x = 1, due to the factor in the numerator. The factor in the denominator, (x - 2), will give us the vertical asymptote at x = 2. And lastly, since the highest degree of x is the same in both the numerator and denominator, and the coefficient of x in the numerator is 1 while the coefficient of x in the denominator is 2, the horizontal asymptote will be at y = 1/2 as x approaches infinity.