Question

Find the domain of the following rational function.
Upper H left parenthesis x right parenthesis equals StartFraction 20 x squared plus x Over x squared plus 10 EndFraction
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Part 1
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
  
enter your response here.
​(Type your answer in interval​ notation.)
B.
There are no restrictions on the domain of​ H(x).

Answer 1

To find the domain of the rational function \( H(x) = \frac{20x^2 + x}{x^2 + 10} \), we need to determine the values of \( x \) for which the function is defined. A rational function is undefined when the denominator is equal to zero.

First, we identify the denominator of the function:

\[
x^2 + 10
\]

Next, we set the denominator equal to zero to find potential restrictions:

\[
x^2 + 10 = 0
\]

We can rearrange this to find:

\[
x^2 = -10
\]

Since \( x^2 \) cannot equal a negative number (as \( x^2 \) is non-negative for all real \( x \)), there are no real solutions to this equation. This means the denominator \( x^2 + 10 \) is never zero for any real number \( x \).

Since the denominator does not equal zero for any real \( x \), there are no restrictions on the domain of \( H(x) \).

Thus, the answer is:

**B. There are no restrictions on the domain of \( H(x) \).**