Question

find the domain of the following rational function.

\\h(x) = \frac{4x^2 + x}{x^2 + 6}\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the domain of h(x) is restricted to
(type your answer in interval notation.)

b. there are no restrictions on the domain of h(x).

Explanation:

Identify the function and domain condition

Using the Rational Functions and Domain Exclusion knowledge points
The given rational function is:

$$ H(x) = \frac{4x^2 + x}{x^2 + 6} $$

To find the domain, we must identify any values of \(x\) that make the denominator equal to zero.

Analyze the denominator

Using the Domain Exclusion knowledge point
We set the denominator equal to zero to find any restricted values:

$$ x^2 + 6 = 0 $$

$$ x^2 = -6 $$

Since the square of any real number \(x\) is always non-negative (\(x^2 \ge 0\)), there is no real number \(x\) that satisfies \(x^2 = -6\).

Determine domain restrictions

Using the Domain Exclusion knowledge point
Because the denominator \(x^2 + 6\) is never zero for any real number \(x\), there are no real values excluded from the domain.
Thus, the domain of \(H(x)\) is all real numbers, meaning there are no restrictions.

Select the correct option

Using the Interval Notation knowledge point
Since there are no restrictions on the domain, we select option B.

Answer:

• (A) The domain of H(x) is restricted to [ ] (Type your answer in interval notation.)
• (B) There are no restrictions on the domain of H(x). (Correct answer)