Question

assignment active identify asymptotes. consider the following equation: $f(x)=\frac{x^2 + 4}{4x^2 - 4x - 8}$ name the vertical asymptote(s). \\(\boldsymbol{\checkmark x = -1}\\) and \\(\boldsymbol{x = 2\checkmark}\\) complete because \\(\circ m < n\\) \\(\circ m = n\\) \\(\circ a_m < b_n\\) \\(\circ a_m = b_n\\) \\(\circ\\) this is where the function is undefined

Explanation:

Step1: Recall vertical asymptote rule

Vertical asymptotes occur where the rational function is undefined, i.e., when the denominator equals 0 (and the numerator does not equal 0 at those points).

Step2: Verify denominator roots

Factor denominator: $4x^2-4x-8=4(x^2-x-2)=4(x+1)(x-2)$. Set to 0: $x=-1, x=2$. Numerator $x^2+4$ is never 0, so these are valid vertical asymptotes.

Step3: Match to reasoning

The vertical asymptotes exist because the function is undefined at these $x$-values.

Answer:

this is where the function is undefined