Question

WILL FAN AND MEDAL AND CHOOSE BEST ANSWER! Write a rational function based on the given description. I NEED HELP WITH RATIONAL FUNCTIONS! Please show work. The question is in the image below.

Answer 1

f(x) = (x² - 7x + 12)/(x³ - 8x² + 20x - 16) = (x - 3)(x - 4)/(x - 2)²(x - 4)

Explanation:

∵ The hole at x = 4

∴ The numerator and denominator both have (x - 4)

∵ Horizontal asymptote at y = 0

∴ The degree of the numerator less than the degree of the denominator

∵ The x-intercept = 3

∴ The numerator has (x -3)

∴ The numerator is (x - 3)(x - 4)

∵ The vertical asymptotes at x = 2

∴ The denominator has (x - 2)²

∴ The denominator is (x - 2)²(x - 4)

[We repeat (x - 2) to make the degree of numerator less than the degree of the denominator]

∴ The numerator is (x² - 7x + 12)

∴ The denominator is (x³ - 8x² + 20x - 16)

∴ f(x) = (x² - 7x + 12)/(x³ - 8x² + 20x - 16) = (x - 3)(x - 4)/(x - 2)²(x - 4)