Question

the functions f and g are defined as follows.
$f(x) = \frac{x + 6}{x^2 + x - 30}$
$g(x) = \frac{x}{x^2 + 16}$
for each function, find the domain.
write each answer as an interval or union of intervals.
domain of f:
domain of g:

Explanation:

Step1: Find undefined points for $f(x)$

Set denominator to 0: $x^2 + x - 30 = 0$
Factor: $(x+6)(x-5)=0$
Solve: $x=-6$ or $x=5$

Step2: Define domain of $f(x)$

Exclude $x=-6$ and $x=5$:
$(-\infty, -6) \cup (-6, 5) \cup (5, \infty)$

Step3: Find undefined points for $g(x)$

Set denominator to 0: $x^2 + 16 = 0$
Solve: $x^2 = -16$, no real solutions

Step4: Define domain of $g(x)$

All real numbers are allowed:
$(-\infty, \infty)$

Answer:

Domain of $f$: $(-\infty, -6) \cup (-6, 5) \cup (5, \infty)$
Domain of $g$: $(-\infty, \infty)$