Find the domain of the following rational function. $$ R(x)=\frac{6\left(x^{2}-5 x-14 ight)}{7\left(x^{2}-49 ight)} $$ Select the correct choice below and fill in any answer boxes within your choice. A. The domain of $$ R(x) $$ is $$ \{x \mid \square\} $$. (Type an inequality in the form $$ x
eq $$. Use integers or fractions for any numbers in the expressior B. The domain of $$ R(x) $$ is the set of all real numbers.

Question

Find the domain of the following rational function. $$ R(x)=\frac{6\left(x^{2}-5 x-14ight)}{7\left(x^{2}-49ight)} $$ Select the correct choice below and fill in any answer boxes within your choice. A. The domain of $$ R(x) $$ is $$ \{x \mid \square\} $$. (Type an inequality in the form $$ x 
eq $$. Use integers or fractions for any numbers in the expressior B. The domain of $$ R(x) $$ is the set of all real numbers.

UpStudy AI · Accepted Answer

1. Identify the function given:
   $$
   R(x)=\frac{6\left(x^{2}-5x-14\right)}{7\left(x^{2}-49\right)}
   $$

2. The domain of a rational function includes all real numbers except where the denominator is zero. Therefore, we need to find when:
   $$
   7\left(x^{2}-49\right) = 0.
   $$

3. Since $$ 7 \neq 0 $$, set the quadratic factor equal to zero:
   $$
   x^{2}-49=0.
   $$

4. Factor the difference of squares:
   $$
   x^{2}-49=(x-7)(x+7)=0.
   $$

5. Solve for $$ x $$ by setting each factor equal to zero:
   $$
   x-7=0 \quad \Longrightarrow \quad x=7,
   $$
   $$
   x+7=0 \quad \Longrightarrow \quad x=-7.
   $$

6. Since $$ R(x) $$ is undefined when the denominator is zero, exclude $$ x=7 $$ and $$ x=-7 $$ from the domain.

7. Therefore, the domain of $$ R(x) $$ is:
   $$
   \{x \mid x \neq -7, \, x \neq 7\}.
   $$

Select choice A.
The domain of $$ R(x) $$ is $$ \{x \mid x \neq -7, \, x \neq 7\} $$.

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