Question

Write an equation for a rational function with the given characteristics: Vertical asymptotes at x = -4 and x = -1, x-intercepts at (1,0) and (5,0), y-intercept at (0,7).

a) f(x) = (7x) / ((x + 4)(x + 1))
b) f(x) = (7x) / ((x - 4)(x - 1))
c) f(x) = (7x) / ((x + 4)(x - 1))
d) f(x) = (7x) / ((x - 4)(x + 1))

Answer 1

The equation for the rational function with the given characteristics is f(x) = (7x) / ((x + 4)(x + 1)).

Step-by-step explanation:

The equation for a rational function with the given characteristics is f(x) = (7x) / ((x + 4)(x + 1)) (option a).

To find the equation, we need to consider the x-intercepts, y-intercept, and vertical asymptotes.

Since the x-intercepts are (1,0) and (5,0), the factors (x-1) and (x-5) should be in the denominator. The y-intercept at (0,7) means that the function passes through (0,7), resulting in the constant term 7. Lastly, the vertical asymptotes at x = -4 and x = -1 correspond to the factors (x+4) and (x+1) in the denominator. Combining all these factors, we get the equation f(x) = (7x) / ((x + 4)(x + 1)).