Question
Find the domain of the following
F(x)=under root x /x^2-x-6
F(x)=under root x /x^2-x-6
Answers
GPT 3.5
To find the domain of a function, we need to consider any restrictions on the input variable, x, that would cause the function to be undefined.
In this case, the function F(x) = √(x) / (x^2 - x - 6), we need to check for two potential issues:
1. The square root must have a non-negative value, so x must be greater than or equal to 0.
2. The denominator (x^2 - x - 6) cannot be equal to zero since division by zero is undefined. We can solve the quadratic equation x^2 - x - 6 = 0 to find the values that would make the denominator zero.
Using factoring or the quadratic formula, we can find that the roots of x^2 - x - 6 = 0 are x = -2 and x = 3. Therefore, x cannot be equal to -2 or 3.
Taking these restrictions into account, the domain of the function F(x) is all real numbers x such that x ≥ 0 and x is not equal to -2 or 3. In interval notation, the domain can be expressed as [0, ∞) \ {-2, 3}.
In this case, the function F(x) = √(x) / (x^2 - x - 6), we need to check for two potential issues:
1. The square root must have a non-negative value, so x must be greater than or equal to 0.
2. The denominator (x^2 - x - 6) cannot be equal to zero since division by zero is undefined. We can solve the quadratic equation x^2 - x - 6 = 0 to find the values that would make the denominator zero.
Using factoring or the quadratic formula, we can find that the roots of x^2 - x - 6 = 0 are x = -2 and x = 3. Therefore, x cannot be equal to -2 or 3.
Taking these restrictions into account, the domain of the function F(x) is all real numbers x such that x ≥ 0 and x is not equal to -2 or 3. In interval notation, the domain can be expressed as [0, ∞) \ {-2, 3}.