Question

problem. 5 : consider the rational function $f(x) = \frac{x^2 + 2x}{x^2 - 4}$. identify any vertical asymptotes.
$x = 2$
problem. 5.1 : are there any holes? if so, give the coordinates of the hole. if not, enter
one\.
$(?, ?)$
problem. 6 : consider the rational function $f(x) = \frac{x^2 - 9x + 20}{x^2 - 3x + 2}$. identify any vertical asymptotes.
(note: input answers below in increasing values of $x$; ie if your answers were $x = -3$ and $x = 22$, then the left answer box would be $-3$ and the right would be $22$).
$x = 1$
$x = 2$
problem. 6.1 : are there any holes? if so, give the coordinates of the hole. if not, enter
one\.
$(?, ?)$

Explanation:

Problem 5.1

Step1: Factor numerator and denominator

Factor \( f(x)=\frac{x^2 + 2x}{x^2 - 4} \). Numerator: \( x(x + 2) \). Denominator: \( (x - 2)(x + 2) \).

Step2: Cancel common factors

Cancel \( (x + 2) \) (for \( x
eq - 2 \)). The hole occurs at \( x=-2 \) (where the common factor is zero).

Step3: Find y - coordinate of the hole

Substitute \( x = - 2 \) into the simplified function \( \frac{x}{x - 2} \). So \( y=\frac{-2}{-2 - 2}=\frac{-2}{-4}=\frac{1}{2} \).

Step1: Factor numerator and denominator

Factor \( f(x)=\frac{x^2-9x + 20}{x^2-3x + 2} \). Numerator: \( (x - 4)(x - 5) \). Denominator: \( (x - 1)(x - 2) \).

Step2: Check for common factors

There are no common factors between the numerator and the denominator.

Answer:

\((-2, \frac{1}{2})\)

Problem 6.1