Determine the domain of (f/g)(x) when f(x)=1/x and g(x)= sqrt x+8
The function (f/g)(x) represents the quotient of f(x) and g(x). Therefore, to determine the domain of (f/g)(x), we need to consider the domains of both f(x) and g(x) and find where they intersect.
First, let's consider the domain of f(x) = 1/x. Since division by zero is undefined, the denominator cannot be zero. Therefore, x ≠ 0. The domain of f(x) is all real numbers except zero, or (-∞, 0) U (0, ∞).
Next, let's consider the domain of g(x) = √(x + 8). The square root function is defined for non-negative real numbers. Therefore, x + 8 ≥ 0, which means x ≥ -8. The domain of g(x) is all real numbers greater than or equal to -8, or [-8, ∞).
To find the domain of (f/g)(x), we need to find where the domains of f(x) and g(x) intersect. In this case, the domain is the intersection of the two domains.
Therefore, the domain of (f/g)(x) is the set of all real numbers except zero and all real numbers greater than or equal to -8, which can be written as (-∞, -8] U (0, ∞).
To determine the domain of (f/g)(x), where f(x) = 1/x and g(x) = √x + 8, we need to consider two factors:
1. The denominator of g(x) should not be zero.
2. The argument of the square root (√) should not be negative.
Let's analyze each condition separately:
First, the denominator of g(x) is √x + 8. For the denominator to be non-zero, we need to exclude the values of x that make √x + 8 equal to zero. Thus, we solve the equation:
√x + 8 ≠ 0
Subtracting 8 from both sides, we get:
√x ≠ -8
Since the square root (√) of a number cannot be negative, there are no restrictions on the domain based on the denominator.
Next, we consider the argument of the square root (√), which is x. For the argument to be non-negative, we have:
x ≥ 0
Combining both conditions, we have:
x ≥ 0
Therefore, the domain of (f/g)(x) is all real numbers greater than or equal to zero, or [0, ∞).
To determine the domain of (f/g)(x), where f(x) = 1/x and g(x) = sqrt(x+8), we need to consider any restrictions that may occur.
First, let's recall that division by zero is undefined. So, we need to ensure that the denominator, g(x), is not equal to zero.
In this case, g(x) = sqrt(x+8), which means the expression under the square root cannot be negative. Therefore, x+8 must be greater than or equal to zero:
x + 8 ≥ 0
To solve this inequality, we subtract 8 from both sides:
x ≥ -8
This means that x has to be greater than or equal to -8 for the expression g(x) to be defined. Therefore, we have found the domain of g(x) as x ≥ -8.
Now let's consider the domain of f(x) = 1/x. Since division by zero is undefined, x cannot be equal to zero. Therefore, the domain of f(x) is all real numbers except x = 0.
Now we can determine the domain of (f/g)(x) by finding the intersection of the domains of f(x) and g(x). Hence, the domain of (f/g)(x) is all real numbers greater than or equal to -8, except x ≠ 0.
In interval notation, the domain can be written as (-∞, -8] ∪ (-8, 0) ∪ (0, ∞).