Question

Give an example of a rational function that has a horizontal asymptote of y = 1/2 and a vertical asymptote at x = -3

a. f(x) = (x + 1)/(2x + 4)
b. f(x) = (2x - 1)/(x + 3)
c. f(x) = (x - 1)/(2x + 3)
d. f(x) = (3x + 2)/(x - 3)

Answer 1

The correct answer is a. f(x) = (x + 1)/(2x + 4). This function has a horizontal asymptote of y = 1/2 and a vertical asymptote at x = -3.

Step-by-step explanation:

To find a rational function with a horizontal asymptote of y = 1/2 and a vertical asymptote at x = -3, we need to find a function that approaches 1/2 as x approaches positive or negative infinity, and has a vertical asymptote at x = -3. The correct answer is a. f(x) = (x + 1)/(2x + 4). Let's analyze this function:

As x approaches positive or negative infinity, the terms with the highest powers in the numerator and denominator dominate, which are x and 2x, respectively. Therefore, the function approaches y = 1/2 as x approaches positive or negative infinity.

Additionally, the denominator 2x + 4 becomes zero when x = -2, but not when x = -3. Therefore, the function has a vertical asymptote at x = -3.