Question

Write the equation of a rational function g(x) that satisfies ALL of the criteria below.

A. g has an x-intercept at x=5.
B. g has a vertical asymptote at x=-2.
C. g has a horizontal asymptote at y= 2/7 .
D. g has a hole at x=-3.

Answer 1

To find an equation of a rational function that satisfies all the given criteria, we can follow these steps.

Step-by-step explanation:

To find an equation of a rational function that satisfies all the given criteria, we can follow these steps:

1. Since g has an x-intercept at x = 5, we know that (x - 5) is a factor of the numerator.
2. Since g has a vertical asymptote at x = -2, we know that (x + 2) is a factor of the denominator.
3. Since g has a horizontal asymptote at y = 2/7, the degree of the numerator should be equal to the degree of the denominator. In this case, the degree can be 1, so we can choose a constant, say k, as the numerator.
4. Since g has a hole at x = -3, we know that (x + 3) is a factor of both the numerator and the denominator, which cancels out the hole. Therefore, we can rewrite the equation as g(x) = k(x - 5)(x + 3)/(x + 2)(x + 3) = k(x - 5)/(x + 2).