To determine when the expression [tex]\(\frac{p(x)}{q(x)} - \frac{r(x)}{s(x)}\)[/tex] is defined, we need to examine the conditions under which both fractions are defined.
1. Identifying the Requirements:
- For the fraction [tex]\(\frac{p(x)}{q(x)}\)[/tex] to be defined, the denominator [tex]\(q(x)\)[/tex] must not be zero. Thus, we require:
[tex]\[
q(x) \neq 0
\][/tex]
- Similarly, for the fraction [tex]\(\frac{r(x)}{s(x)}\)[/tex] to be defined, the denominator [tex]\(s(x)\)[/tex] must not be zero. Thus, we also require:
[tex]\[
s(x) \neq 0
\][/tex]
2. Combining the Conditions:
- Both fractions [tex]\(\frac{p(x)}{q(x)}\)[/tex] and [tex]\(\frac{r(x)}{s(x)}\)[/tex] must be defined simultaneously. Therefore, we need both of the above conditions to be satisfied at the same time, which gives us the combined requirement:
[tex]\[
q(x) \neq 0 \text{ and } s(x) \neq 0
\][/tex]
3. Conclusion:
- Thus, the expression [tex]\(\frac{p(x)}{q(x)} - \frac{r(x)}{s(x)}\)[/tex] is defined when [tex]\(q(x) \neq 0\)[/tex] and [tex]\(s(x) \neq 0\)[/tex].
In summary, the expression [tex]\(\frac{p(x)}{q(x)} - \frac{r(x)}{s(x)}\)[/tex] is defined if and only if [tex]\(q(x)\)[/tex] is not equal to zero for all [tex]\(x\)[/tex] in the domain and [tex]\(s(x)\)[/tex] is not equal to zero for all [tex]\(x\)[/tex] in the domain.
