What is the domain of f (x) = sqr root of x^2 - 5x - 14?
You need to use parentheses to clarify what number you are taking the square root of. I assume it is
x^2 - 5x - 14.
It would have to be a positive number or zero. That will help you establish the domain.
x^2 -5x -14 is zero when (x-7)(x+2) = 0, which means at x = -2 and x = 7.
Between x=-2 and +7, it is negative.
The domain is therefore x<-2 and x>+5
To determine the domain of the function f(x) = √(x^2 - 5x - 14), we need to consider the values of x for which the function is defined.
Since the function involves a square root (√), the expression under the square root (√(x^2 - 5x - 14)) must be non-negative. We solve the inequality:
x^2 - 5x - 14 ≥ 0
Next, we find the roots of the quadratic equation x^2 - 5x - 14 = 0 by factoring, completing the square, or using the quadratic formula. The roots are x = -2 and x = 7.
Now, we analyze the sign of the quadratic expression in the inequality x^2 - 5x - 14 ≥ 0 for different intervals.
1. For x < -2: Choose a value between -2 and test it in the inequality, like x = -3:
(-3)^2 - 5(-3) - 14 = 9 + 15 - 14 = 10 > 0
The quadratic expression is positive, so this interval is not part of the domain.
2. For -2 ≤ x ≤ 7: Choose a value between -2 and 7, like x = 0:
(0)^2 - 5(0) - 14 = -14 < 0
The quadratic expression is negative, so this interval is not part of the domain.
3. For x > 7: Choose a value greater than 7 and test it in the inequality, like x = 8:
(8)^2 - 5(8) - 14 = 64 - 40 - 14 = 10 > 0
The quadratic expression is positive, so this interval is part of the domain.
Based on this analysis, the domain of f(x) = √(x^2 - 5x - 14) is x > 7.