Find the domain of the following function and give your answer in interval notation
F(x)=root x/x^2+7x+12
To determine the domain of the function F(x) = √(x)/(x^2 + 7x + 12), we need to find the values of x that make the function undefined.
First, let's consider the denominator x^2 + 7x + 12. Since dividing by zero is undefined, we need to find the values of x that make the denominator equal to zero.
Factorizing the denominator:
x^2 + 7x + 12 = (x + 3)(x + 4)
Setting each factor equal to zero:
x + 3 = 0 --> x = -3
x + 4 = 0 --> x = -4
Therefore, the function F(x) is undefined when x = -3 or x = -4, as these values would make the denominator zero.
The domain of the function F(x) is all real numbers except -3 and -4. In interval notation, this can be expressed as:
(-∞, -4) U (-4, -3) U (-3, ∞)
To find the domain of the function F(x) = sqrt(x) / (x^2 + 7x + 12), we need to determine the values of x that make the function undefined.
First, note that the function has a square root, which means that the value inside the square root (x) must be non-negative. Therefore, x ≥ 0.
Next, observe that the denominator of the function is a quadratic expression (x^2 + 7x + 12). To ensure that the function is defined, the denominator cannot be equal to zero. We can find the values of x that make the denominator zero by factoring the quadratic expression:
x^2 + 7x + 12 = 0
(x + 3)(x + 4) = 0
Setting each factor equal to zero gives x + 3 = 0 → x = -3 and x + 4 = 0 → x = -4.
So, the function F(x) is undefined when x equals -3 or -4.
In interval notation, the domain of F(x) is [0, -4) U (-4, -3) U (-3, ∞).