Find the domain of srt(6-x-x^2. Thank you!
since you can't get the square root of a negative number find when 6-x-x^2 < 0
It's a quadratic that you can solve. Plot it and see where the graph is above the x-axis. The domain is the possible x-values that give y-values greater than or equal to 0.
GanonTEK is correct. If you don't have ready access to a graphing utility, just use what you know about parabolas. Since this is one which opens downward, it is positive between the roots, where it crosses the x-axis.
To find the domain of the function √(6 - x - x²), we need to consider two things: the radicand (the expression under the square root) and any restrictions on the square root function.
First, we look at the radicand, which is 6 - x - x². For this expression to be defined, the radicand cannot be negative, as the square root of a negative number is not a real number. So, we set up the inequality:
6 - x - x² ≥ 0
Rearranging this inequality, we have:
x² + x - 6 ≤ 0
Now, let's factor the quadratic equation:
(x + 3)(x - 2) ≤ 0
Next, we find the critical points by setting each factor equal to zero:
x + 3 = 0 => x = -3
x - 2 = 0 => x = 2
Now we plot these critical points on a number line. This divides the number line into three intervals: (-∞, -3), (-3, 2), and (2, +∞).
For each interval, we choose a test point and substitute it into the inequality to determine whether it is true or false. To simplify, we can simply substitute 0 into the inequality.
Testing the interval (-∞, -3), we have:
(0 + 3)(0 - 2) ≤ 0
3*(-2) ≤ 0
-6 ≤ 0
Since -6 is less than or equal to zero, the inequality is true in this interval.
Testing the interval (-3, 2), we have:
(0 + 3)(0 - 2) ≤ 0
3*(-2) ≤ 0
-6 ≤ 0
Same as before, -6 is less than or equal to zero, so the inequality is true in this interval.
Testing the interval (2, +∞), we have:
(0 + 3)(0 - 2) ≤ 0
3*(-2) ≤ 0
-6 ≤ 0
Again, -6 is less than or equal to zero, so the inequality is true in this interval.
Since the inequality is true for all three intervals, the solution to the inequality is:
(-∞, -3] U [2, +∞)
Therefore, the domain of the function √(6 - x - x²) is (-∞, -3] U [2, +∞).