Find the domain of the following rational function.
Upper H left parenthesis x right parenthesis equals StartFraction 4 x squared plus x Over x squared plus 7 EndFraction

The rational function you provided is:

\[
H(x) = \frac{4x^2 + x}{x^2 + 7}
\]

To find the domain of this function, we need to determine where the function is defined. A rational function is undefined wherever the denominator is zero. So, we set the denominator equal to zero and solve for \( x \):

\[
x^2 + 7 = 0
\]

\[
x^2 = -7
\]

Since the square of a real number can never be negative, there are no real solutions to this equation. This means the denominator is never zero for any real number \( x \).

Thus, the domain of the function is all real numbers, or in interval notation:

\[
\boxed{(-\infty, \infty)}
\]

Let me know if you'd like more details or have any questions.

Here are 5 related questions:
1. What is the range of the function \( H(x) \)?
2. How do you find the asymptotes of a rational function?
3. What happens to \( H(x) \) as \( x \to \infty \)?
4. Can rational functions have removable discontinuities?
5. How would the domain change if the denominator were \( x^2 - 7 \) instead?

**Tip**: Always check both the numerator and the denominator when analyzing rational functions, as they help determine key properties like domain and asymptotes.

Find the domain of the following rational function. H(x) = (4x^2 + x) / (x^2 + 7)

Learn how to find the domain of the rational function H(x) = (4x^2 + x) / (x^2 + 7). This solution explains the key steps to determine where the function is undefined and concludes that the domain is all real numbers. Suitable for high school students in Grades 9-12.

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