Question

find all vertical asymptotes of the following function.
$f(x) = \dfrac{x^2 - 2x - 15}{3x - 15}$

Explanation:

Step1: Identify denominator zero

To find vertical asymptotes, set the denominator of \( f(x)=\frac{x^{2}-2x - 15}{3x-15} \) equal to zero.
\( 3x - 15=0 \)

Step2: Solve for x

Solve the equation \( 3x - 15 = 0 \). Add 15 to both sides: \( 3x=15 \). Divide by 3: \( x = 5 \).

Step3: Check numerator at x=5

Check if the numerator is non - zero at \( x = 5 \). Substitute \( x = 5 \) into the numerator \( x^{2}-2x - 15 \): \( 5^{2}-2\times5 - 15=25 - 10 - 15 = 0 \). Since both numerator and denominator are zero at \( x = 5 \), we need to simplify the function.

Step4: Simplify the function

Factor numerator and denominator. Numerator: \( x^{2}-2x - 15=(x - 5)(x+3) \). Denominator: \( 3x - 15 = 3(x - 5) \). Then \( f(x)=\frac{(x - 5)(x + 3)}{3(x - 5)}=\frac{x + 3}{3},x
eq5 \). The function has a hole at \( x = 5 \) (a removable discontinuity) and no vertical asymptotes.

Answer:

The function \( f(x)=\frac{x^{2}-2x - 15}{3x - 15} \) has no vertical asymptotes (it has a hole at \( x = 5 \)).