Find the domain of f(x) = 6/(x - 15) + 6/(x + 12), giving the answer in interval notation.

Step-by-step solution

1. 1

   Identify what can go wrong. The expression is a sum of two rational terms with no radicals or logarithms, so the only restriction is division by zero. Each denominator must be checked separately — a value that kills either one kills the whole function.

2. 2

   Solve each denominator equal to zero.

   $x - 15 = 0 \ \Longrightarrow\ x = 15, \qquad x + 12 = 0 \ \Longrightarrow\ x = -12$

3. 3

   Resist the temptation to combine the fractions first. Adding them gives $\dfrac{6(x+12) + 6(x-15)}{(x-15)(x+12)} = \dfrac{12x - 18}{(x-15)(x+12)}$. The combined denominator has the same two zeros, so nothing changes here — but if a numerator factor had cancelled one of them, the original function would still be undefined there. Always read restrictions off the original form.

4. 4

   Remove the two points from the real line. Deleting $-12$ and $15$ from $\mathbb{R}$ leaves three open intervals, ordered from left to right:

   $(-\infty,\, -12) \cup (-12,\, 15) \cup (15,\, \infty)$

5. 5

   Check a value in each interval. At $x = -20$: $\tfrac{6}{-35} + \tfrac{6}{-8} = -0.9214$, defined. At $x = 0$: $\tfrac{6}{-15} + \tfrac{6}{12} = 0.1$, defined. At $x = 20$: $\tfrac{6}{5} + \tfrac{6}{32} = 1.3875$, defined. At $x = 15$ and $x = -12$ the function has vertical asymptotes.