Question

Question
Which of the following is a rational expression whose domain has a restriction of x≠−1
?(1 point)
Responses

x+1x3
Start Fraction x plus 1 over x cubed End Fraction

6x3+60x
Start Fraction 6 x cubed plus 60 over x End Fraction

x√2x+2
Start Fraction Start Root x End Root over 2 x plus 2 End Fraction

x2+6x+56x+6
Start Fraction x squared plus 6 x plus 5 over 6 x plus 6 End Fraction
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Answer 1

To determine which of the given rational expressions has a domain restriction of \( x \neq -1 \), we need to analyze each expression for potential values of \( x \) that would make the denominator equal to zero (as these are the values that cannot be included in the domain).

1. **Expression**: \( \frac{x + 1}{x^3} \)
- Denominator: \( x^3 \)
- Restrictions: \( x^3 = 0 \) when \( x = 0 \). No restriction at \( x = -1 \).

2. **Expression**: \( \frac{6x^3 + 60}{x} \)
- Denominator: \( x \)
- Restrictions: \( x = 0 \). No restriction at \( x = -1 \).

3. **Expression**: \( \frac{\sqrt{x}}{2x + 2} \)
- Denominator: \( 2x + 2 \)
- Restrictions: \( 2x + 2 = 0 \) when \( x = -1 \). This means that \( x = -1 \) is restricted.

4. **Expression**: \( \frac{x^2 + 6x + 5}{6x + 6} \)
- Denominator: \( 6x + 6 \)
- Restrictions: \( 6x + 6 = 0 \) when \( x = -1 \). This means that \( x = -1 \) is restricted.

From the analysis, expressions 3 and 4 have a restriction at \( x = -1 \).

However, since the question asks for just one rational expression with the restriction \( x \neq -1 \), **the correct answers would be**:
- \( \frac{\sqrt{x}}{2x + 2} \) (Expression 3)
- \( \frac{x^2 + 6x + 5}{6x + 6} \) (Expression 4)

But since only one response is typically expected, **if you need to select just one,** you would likely state either one that specifically mentions \( x \neq -1 \). If you have to select one, I would recommend **Expression 4** as a comprehensive rational expression, demonstrating both a quadratic numerator and a linear denominator.