Which of the following is a rational expression whose domain has a restriction of x≠−1 ?(1 point)
1. x+1/x^3
2. x^2+6x+5/6x+6
3. √x/2x+2
4. 6x^3+60/x
To determine which of the given expressions has a domain restriction of x ≠ -1, we need to identify any denominators that might become zero if x equals -1. Recall that division by zero is undefined.
Let's go through each option and analyze their denominators:
1. x + 1 / x^3
The denominator here is x^3. Since the denominator does not contain x - 1 as a factor, there is no domain restriction. Hence, option 1 does not have a domain restriction of x ≠ -1.
2. (x^2 + 6x + 5) / (6x + 6)
The denominator here is 6x + 6. To check if it can be zero:
6x + 6 = 0
Dividing both sides by 6, we get:
x + 1 = 0
x = -1
The denominator can indeed become zero when x = -1. Therefore, option 2 has a domain restriction of x ≠ -1.
3. √x / (2x + 2)
The denominator here is 2x + 2. To check for domain restriction, we set it equal to zero:
2x + 2 = 0
Dividing both sides by 2, we have:
x + 1 = 0
x = -1
The denominator can also become zero when x = -1. Hence, option 3 has a domain restriction of x ≠ -1.
4. (6x^3 + 60) / x
The denominator here is x. Since the denominator does not contain x - 1 as a factor, there is no domain restriction. Thus, option 4 does not have a domain restriction of x ≠ -1.
In summary, the rational expressions with a domain restriction of x ≠ -1 are options 2 and 3.
2. x^2+6x+5/6x+6
To identify the rational expression with a domain restriction of x ≠ -1, we need to check each option and see if any of them have a denominator that becomes zero when x = -1.
1. x + 1/x^3:
The denominator is x^3, which is never equal to zero. Therefore, this rational expression does not have a domain restriction of x ≠ -1.
2. (x^2 + 6x + 5) / (6x + 6):
The denominator is 6x + 6. When x = -1, the denominator becomes zero:
6(-1) + 6 = -6 + 6 = 0
Thus, this rational expression has a domain restriction of x ≠ -1.
3. √x / (2x + 2):
The denominator is 2x + 2. When x = -1, the denominator becomes zero:
2(-1) + 2 = -2 + 2 = 0
Therefore, this rational expression has a domain restriction of x ≠ -1.
4. (6x^3 + 60) / x:
The denominator is x. When x = -1, the denominator becomes zero.
Therefore, this rational expression has a domain restriction of x ≠ -1.
Based on the analysis, options 2, 3, and 4 all have a domain restriction of x ≠ -1.