Question

identify the vertical asymptotes of each. 9) $f(x)=\frac{x + 3}{x^{2}+2x - 3}$ 10) $f(x)=\frac{-3x^{2}-3x}{x^{3}-4x}$

Explanation:

Step1: Factor denominator (Q9)

$x^2 + 2x - 3 = (x+3)(x-1)$

Step2: Simplify function (Q9)

$f(x)=\frac{x+3}{(x+3)(x-1)}=\frac{1}{x-1}$ (for $x
eq-3$)

Step3: Find undefined x (Q9)

$x-1=0 \implies x=1$

Step4: Factor numerator/denominator (Q10)

Numerator: $-3x^2-3x=-3x(x+1)$
Denominator: $x^3-4x=x(x^2-4)=x(x-2)(x+2)$

Step5: Simplify function (Q10)

$f(x)=\frac{-3x(x+1)}{x(x-2)(x+2)}=\frac{-3(x+1)}{(x-2)(x+2)}$ (for $x
eq0$)

Step6: Find undefined x (Q10)

$(x-2)(x+2)=0 \implies x=2, x=-2$

Answer:

1. Vertical asymptote: $x=1$
2. Vertical asymptotes: $x=2$ and $x=-2$