Find the domain of f(x)=√(6+x-x²)
You got it exactly backwards. Since the parabola opens downward, f(x) ≥ 0 between the roots. That is, the domain for f(x) is [-2,3].
Apologies for the mistake. You are correct. The domain of f(x) = √(6+x-x²) is [-2, 3]. Thank you for catching the error.
To find the domain of the function f(x) = √(6+x-x²), we need to determine the values of x for which the function is defined.
The square root function (√) is defined only for non-negative real numbers. Therefore, the expression inside the square root, 6+x-x², must be non-negative.
To find the solutions to the inequality 6+x-x² ≥ 0, we can first factor the quadratic expression: -x² + x + 6.
(-x + 3)(x + 2) ≥ 0.
Now, we can use the fact that the product of two factors is non-negative if and only if both factors have the same sign or one of the factors is zero.
Case 1: -x + 3 ≥ 0 and x + 2 ≥ 0.
Solving these inequalities, we get x ≤ 3 and x ≥ -2. The intersection of these two solution sets is x ≥ -2.
Case 2: -x + 3 ≤ 0 and x + 2 ≤ 0.
Solving these inequalities, we get x ≥ 3 and x ≤ -2. The intersection of these two solution sets is x ≤ -2.
Therefore, the domain of f(x) is x ≤ -2 or x ≥ 3.
To find the domain of f(x) = √(6+x-x²), we need to consider the values of x that make the expression under the square root non-negative.
The expression under the square root, 6+x-x², can be rewritten as -x² + x + 6.
Since the square root of a negative number (i.e., when the expression under the square root is negative) is not a real number, we want to find x values that make -x² + x + 6 ≥ 0.
To solve this inequality, we can factor the expression: -x² + x + 6 = (x - 3)(-x - 2).
Setting each factor equal to zero, we have x - 3 = 0 and -x - 2 = 0.
Solving for x, we find x = 3 and x = -2.
Now we can create an interval chart to determine the sign of the expression (-x² + x + 6) for different x values:
----------------------
(-∞, -2) | +
----------------------
(-2, 3) | -
----------------------
(3, ∞) | +
----------------------
From the interval chart, we can see that the expression (-x² + x + 6) is non-negative for x ≤ -2 and x ≥ 3.
Therefore, the domain of f(x) = √(6+x-x²) is (-∞, -2] U [3, ∞).