Find the domain of the function f(x) = 2 / (x^2 - 12x + 35). Give your answer in interval notation.

Step-by-step solution

1. 1

   Identify the only restriction a rational function has. The numerator $2$ is harmless, and polynomials are defined for every real number, so the domain is everything except the zeros of the denominator. There is no square root and no logarithm here, so division by zero is the sole hazard. Set up the factoring rather than the quadratic formula, since the numbers are friendly.

2. 2

   Factor the denominator. Look for two numbers multiplying to $35$ and adding to $-12$; both must be negative, and $-5$ and $-7$ work:

   $x^{2}-12x+35=(x-5)(x-7)$

   Expanding back gives $x^{2}-7x-5x+35=x^{2}-12x+35$, so the factoring is correct.

3. 3

   Solve for the excluded values. A product is zero exactly when a factor is zero:

   $x-5=0\ \Rightarrow\ x=5,\qquad x-7=0\ \Rightarrow\ x=7$

   At each of these the denominator vanishes while the numerator stays $2$, so $f$ has a vertical asymptote there rather than a removable hole.

4. 4

   Remove those two points from the real line. Deleting two isolated points from $\mathbb{R}$ splits it into three intervals:

   $(-\infty,5)\cup(5,7)\cup(7,\infty)$

   All four endpoints use parentheses — $5$ and $7$ because they are excluded, and $\pm\infty$ because they are not numbers.

5. 5

   Spot-check either side of an excluded value. At $x=6$ the denominator is $36-72+35=-1$, so $f(6)=-2$, a legitimate value inside the middle interval. At $x=5$ the denominator is $25-60+35=0$, so $f(5)$ is undefined, exactly as the answer says.