Final answer:
To construct a rational function with the specified characteristics, we form the denominator to set the vertical asymptotes, include a canceled factor in the numerator for the hole, choose the leading coefficient for the horizontal asymptote, and adjust the numerator for the y-intercept without adding x-intercepts, resulting in the function f(x) = 3(x - 3)(x + 2)/(x - 2)(x + 4).
Step-by-step explanation:
To create a rational function with the given characteristics (only vertical asymptotes at x = 2 and x = -4, horizontal asymptote at y = 3, a hole at (3,21), and a y-intercept of 3, with no x-intercepts), you can start by forming the denominator to set the vertical asymptotes. The function must factor to (x - 2) and (x + 4) in the denominator:
f(x) = \frac{N(x)}{(x - 2)(x + 4)}
To ensure a hole at (3,21), the numerator must contain a factor of (x - 3), but it should be canceled out to not affect the x-intercepts:
f(x) = \frac{(x - 3)P(x)}{(x - 2)(x + 4)}
To have the horizontal asymptote at y = 3, the leading coefficients of the numerator and denominator should be equal when the function is simplified, meaning we need a leading coefficient of 3 in both the numerator and denominator:
f(x) = \frac{3(x - 3)(x + a)}{(x - 2)(x + 4)}
Given that the y-intercept is 3 (which implies that f(0) = 3), we must solve for 'a' by substituting x with 0:
f(0) = \frac{3(0 - 3)(0 + a)}{(0 - 2)(0 + 4)} = 3
Solve for 'a' and you get a = 2. Now you have enough information to write the function knowing that there are no x-intercepts (P(x) should not change the sign of the product). A possible function can be:
f(x) = \frac{3(x - 3)(x + 2)}{(x - 2)(x + 4)}
This function now satisfies all the criteria given.