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

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

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

6x3+60x

Answer 1

To determine which rational expression has a domain restriction of \( x \neq -1 \), we need to analyze each option to see if substituting \( x = -1 \) causes any discontinuities (like division by zero) or is otherwise undefined.

1. **\( \frac{x+1}{x^3} \)**:
- The denominator \( x^3 \) is zero when \( x = 0 \) (which is a restriction), but not at \( x = -1 \).
- **Domain: all \( x \) except \( x = 0 \)**.

2. **\( \frac{x^2 + 6x + 5}{6x + 6} \)**:
- The denominator \( 6x + 6 = 6(x + 1) \) is zero when \( x = -1 \).
- **Domain: all \( x \) except \( x = -1 \)**.

3. **\( \frac{\sqrt{x}}{2x + 2} \)**:
- The denominator \( 2x + 2 = 2(x + 1) \) is zero when \( x = -1 \) and the square root is defined only for \( x \geq 0 \).
- **Domain: \( x \geq 0\) except \( x = -1\), but since \( -1 \) is not in the domain anyway, it’s valid.**

4. **\( 6x^3 + 60x \)**:
- This is a polynomial and is defined for all \( x \). There are no restrictions.
- **Domain: all \( x \)**.

Out of these options, the only rational expression whose domain has a restriction of \( x \neq -1 \) is:

**\( \frac{x^2 + 6x + 5}{6x + 6} \)**.