Simplify f(x) = (-x² - 289)/x and state its domain.

Step-by-step solution

1. 1

   Check whether the numerator factors first. $-x^2-289 = -(x^2+289)$. Since $x^2+289$ is a sum of squares it has no real factors, so nothing cancels wholesale - splitting the fraction is the only simplification available.

2. 2

   Split the single fraction into two. Division distributes over the terms of the numerator: $\frac{-x^2-289}{x} = \frac{-x^2}{x} - \frac{289}{x}.$ This is legal because both pieces keep the same denominator $x$.

3. 3

   Cancel one factor of x in the first term. $\dfrac{-x^2}{x} = \dfrac{-x\cdot x}{x} = -x$, valid because we are already assuming $x \neq 0$. The second term has no $x$ in its numerator, so it stays as $-\dfrac{289}{x}$.

4. 4

   Assemble the simplified form. $f(x) = -x - \frac{289}{x} = -\left(x + \frac{289}{x}\right).$ The bracketed form makes the structure obvious: $f$ is $-1$ times a number-plus-its-reciprocal-multiple.

5. 5

   State the domain. The original denominator is $x$, so $x=0$ must be excluded; the domain is $x \in (-\infty,0)\cup(0,\infty)$. Note that the simplified form has the same restriction, so no hole was created or destroyed.

6. 6

   Read off the structure as a bonus. Since $f(-x) = x + \dfrac{289}{x} = -f(x)$, the function is odd, and $x = 0$ is a vertical asymptote while $y = -x$ is a slant asymptote.