Question

the functions f and g are defined as follows.
$f(x) = \frac{x^2}{x - 5}$
$g(x) = \frac{x - 8}{x^2 - 17x + 72}$
for each function, find the domain.
write each answer as an interval or union of intervals.

Explanation:

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

Set denominator of $f(x)$ to 0:
$x-5=0$
Solve for $x$: $x=5$

Step2: Define domain of $f(x)$

All real numbers except $x=5$, so:
$(-\infty,5) \cup (5,\infty)$

Step3: Factor denominator of $g(x)$

Factor $x^2-17x+72$:
$x^2-17x+72=(x-8)(x-9)$

Step4: Find undefined points for $g(x)$

Set factored denominator to 0:
$(x-8)(x-9)=0$
Solve for $x$: $x=8$ and $x=9$

Step5: Define domain of $g(x)$

All real numbers except $x=8$ and $x=9$, so:
$(-\infty,8) \cup (8,9) \cup (9,\infty)$

Answer:

Domain of $f(x)$: $(-\infty,5) \cup (5,\infty)$
Domain of $g(x)$: $(-\infty,8) \cup (8,9) \cup (9,\infty)$