Final answer:
The domain of the function f(x) = √(2x² - 5x - 12) is x ≤ -3/2 or x ≥ 4.
Step-by-step explanation:
The domain of the radical function f(x) = √(2x² - 5x - 12) can be determined by finding the values of x that make the expression inside the square root non-negative. This means the expression 2x² - 5x - 12 must be greater than or equal to zero.
To solve this inequality, we can factor the quadratic and find the x-values that satisfy the inequality. The factored form of the quadratic is (2x + 3)(x - 4). We set each factor equal to zero and solve for x to find the critical points.
- (2x + 3) = 0, x = -3/2
- (x - 4) = 0, x = 4
The domain of the function is the set of all x-values that satisfy the inequality. We test intervals between the critical points and use test values to determine the sign of the expression. The domain is x ≤ -3/2 or x ≥ 4.