Question

Find the domain of f(x)=√(6+x-x²)

Answer 1

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, ∞).

Answer 2

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].

Answer 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.