Question

find all vertical asymptotes of the following function.

$f(x)=\frac{25x^{2}-64}{15x + 24}$

Explanation:

Step1: Factor numerator and denominator

Numerator: \(25x^{2}-64=(5x)^{2}-8^{2}=(5x - 8)(5x+8)\) (using \(a^{2}-b^{2}=(a - b)(a + b)\)).
Denominator: \(15x+24 = 3(5x+8)\)
So \(f(x)=\frac{(5x - 8)(5x+8)}{3(5x+8)}\)

Step2: Simplify the function

Cancel out the common factor \((5x + 8)\) (for \(x
eq-\frac{8}{5}\)), we get \(f(x)=\frac{5x - 8}{3}\), \(x
eq-\frac{8}{5}\)

Step3: Find the vertical asymptote

A vertical asymptote occurs where the denominator of the original (unsimplified) rational function is zero (after canceling non - zero common factors).
Set the denominator of the original function \(15x+24 = 0\)
Solve for \(x\): \(15x=-24\), so \(x=-\frac{24}{15}=-\frac{8}{5}\)

Answer:

The vertical asymptote is \(x =-\frac{8}{5}\)