Question

drag the tiles to the correct boxes. not all tiles will be used. what are the domain and the range of function f? f(x) = (x - 6)/(x² - 3x - 18) the blue tiles are: (-∞, 0) ∪ (0, ∞), (-∞, -3) ∪ (-3, ∞), (-∞, -6) ∪ (-6, 3) ∪ (3, ∞), (-∞, 6) ∪ (6, ∞), (-∞, -3) ∪ (-3, 6) ∪ (6, ∞), (-∞, 0) ∪ (0, 1/9) ∪ (1/9, ∞) there are two boxes: domain and range, each with an arrow pointing to a dashed box for the answer.

Explanation:

Step1: Find domain (denominator ≠0)

Factor denominator: $x^2-3x-18=(x-6)(x+3)$
Set $(x-6)(x+3)
eq0$, so $x
eq6$ and $x
eq-3$.
Domain: $(-\infty, -3) \cup (-3, 6) \cup (6, \infty)$

Step2: Simplify function to find range

$f(x)=\frac{x-6}{(x-6)(x+3)}=\frac{1}{x+3}$ (for $x
eq6$)
Let $y=\frac{1}{x+3}$, solve for $x$: $x=\frac{1}{y}-3$
$y
eq0$, and when $x=6$, $y=\frac{1}{6+3}=\frac{1}{9}$, so $y
eq\frac{1}{9}$.
Range: $(-\infty, 0) \cup (0, \frac{1}{9}) \cup (\frac{1}{9}, \infty)$

Answer:

domain: $(-\infty, -3) \cup (-3, 6) \cup (6, \infty)$
range: $(-\infty, 0) \cup (0, \frac{1}{9}) \cup (\frac{1}{9}, \infty)$