Question

Identify the holes, vertical asymptotes and horizontal asymptotes of f(x)=x²+7x+12/-x²-3x+4

Answer 1

Answer:

• hole: x = -4
• vertical asymptote: x = 1
• horizontal asymptote: y = -1

Step-by-step explanation:

The function factors as ...

... f(x) = -(x+3)(x+4)/((x-1)(x+4)) = -(x+3)/(x-1) . . . . x ≠ -4

and the quotient is ...

... f(x) = -(1 +4/(x-1))

... f(x) = -1 -4/(x-1) . . . . . x ≠ -4

This has a hole at x = -4 (where the denominator factor cancels the numerator factor), a vertical asymptote at x=1 (where the denominator is zero), and a horizontal asymptote at y = -1 (the value of the function for large values of x).