Question

the functions f and g are defined as follows.
$f(x) = \frac{x + 8}{x^2 - 16x + 64}$
$g(x) = \frac{x - 7}{x^2 - 49}$
for each function, find the domain.
write each answer as an interval or union of intervals.
domain of f:
domain of g:

Explanation:

Step1: Factor denominator of $f(x)$

$x^2-16x+64=(x-8)^2$

Step2: Find undefined point for $f(x)$

Set $(x-8)^2=0 \implies x=8$

Step3: Define domain of $f(x)$

All real numbers except $x=8$

Step4: Factor denominator of $g(x)$

$x^2-49=(x-7)(x+7)$

Step5: Find undefined points for $g(x)$

Set $(x-7)(x+7)=0 \implies x=7, x=-7$

Step6: Define domain of $g(x)$

All real numbers except $x=7$ and $x=-7$

Answer:

Domain of $f$: $(-\infty,8) \cup (8,\infty)$
Domain of $g$: $(-\infty,-7) \cup (-7,7) \cup (7,\infty)$