Question: Find a rational function that satisfies the given conditions. Vertical asymptotes \( x = -3 \),…

Find a rational function that satisfies the given conditions.

Vertical asymptotes \( x = -3 \), \( x = 6 \)

Solution

We know that vertical asymptotes occur where the denominator is zero. Therefore, the denominator must have factors \[ (x + 3)(x - 6) \] To ensure there are vertical asymptotes at \( x = -3 \) and \( x = 6 \), the denominator is set as above. The numerator can be any polynomial that does not cancel these factors. For simplicity, we can choose the numerator to be 1. Thus, one possible rational function is \[ f(x) = \frac{1}{(x + 3)(x - 6)} \]