Find the Asymptotes f(x)=((x-2)(x-4))/((x-4)(x+1))

Problem

ƒ(x)=((x−2)*(x−4))/((x−4)*(x+1))

Solution

1. Identify the domain by finding values that make the denominator zero.

x−4=0⇒x=4

x+1=0⇒x=−1

The domain is all real numbers except x=4 and x=−1

2. Simplify the function by canceling common factors in the numerator and denominator.

ƒ(x)=(x−2)/(x+1),x≠4

3. Determine vertical asymptotes by finding where the simplified denominator is zero.

x+1=0⇒x=−1

4. Identify holes (removable discontinuities) where factors were canceled.
   Since (x−4) was canceled, there is a hole at x=4

5. Determine horizontal asymptotes by comparing the degrees of the numerator and denominator.
   The degrees are equal (both are 1 in the simplified form or 2 in the original form).

y=1/1=1

Final Answer

Vertical Asymptote: *x=−1, Horizontal Asymptote: *y=1

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