Find the domain of the function f(x) = 1 / (8/(x − 5) − 2) and write it in interval notation.

Step-by-step solution

1. 1

   Count the denominators, not the fractions. A complex fraction has more than one place it can break. This function has two: the inner denominator $x-5$, and the outer denominator $\frac{8}{x-5}-2$. Both must be non-zero, and each gives a separate excluded value.

2. 2

   Restriction 1 — the inner denominator. $\frac{8}{x-5}$ requires

   $x-5\ne 0\quad\Longrightarrow\quad x\ne 5.$

3. 3

   Restriction 2 — the outer denominator. Set the outer denominator equal to zero and solve, then exclude that root:

   $\frac{8}{x-5}-2=0\ \Longrightarrow\ \frac{8}{x-5}=2\ \Longrightarrow\ 8=2(x-5)\ \Longrightarrow\ x=9,$

   so $x\ne 9$.

4. 4

   Why the restrictions must be found before simplifying. Multiplying numerator and denominator by $x-5$ gives the tidy form

   $f(x)=\frac{x-5}{8-2(x-5)}=\frac{x-5}{18-2x}.$

   That form still shows $x\ne 9$ (since $18-2x=0$ at $x=9$), but it has completely lost the $x\ne 5$ restriction. Simplifying first and reading the domain off the simplified expression is the standard way to get this kind of problem wrong.

5. 5

   Write the domain in interval notation. Removing the two isolated points from the real line leaves three open intervals:

   $(-\infty,5)\cup(5,9)\cup(9,\infty).$