Domain of the Expression √(2-2x-x2)

Understanding the Domain in Mathematical Contexts

In the field of mathematics, the domain of a function is the set of all possible input values (x-values) for which the function is defined. When considering the expression (sqrt{2-2x-x^2}), our primary focus is to find the values of x that ensure the expression is real and defined. This involves avoiding any complex numbers and ensuring that the value under the square root is non-negative.

Ensuring a Non-Negative Square Root

The square root of a number is only real when the number is non-negative, i.e., it must satisfy the inequality:

(2-2x-x^2 geq 0)

To solve this inequality, we first tackle it as an equality to find the critical points:

(2-2x-x^2 0)

By rearranging, we have a standard quadratic equation:

(-x^2 - 2x 2 0)

Transforming for Simplicity

To make the quadratic equation simpler, we multiply every term by -1 (this doesn’t change the direction of the inequality since multiplying both sides by a negative number reverses the inequality sign):

(x^2 2x - 2 0)

Solving Using the Quadratic Formula

The general form of a quadratic equation is (ax^2 bx c 0). Given our quadratic equation, we can use the quadratic formula to find the roots:

(x frac{-b pm sqrt{b^2 - 4ac}}{2a})

In our equation, (a 1), (b 2), and (c -2). Plugging in these values, we get:

(x frac{-2 pm sqrt{(2)^2 - 4(1)(-2)}}{2(1)})

(x frac{-2 pm sqrt{4 8}}{2})

(x frac{-2 pm sqrt{12}}{2})

(x frac{-2 pm 2sqrt{3}}{2})

Which simplifies to:

(x -1 pm sqrt{3})

This gives us the roots:

(x -1 sqrt{3})

(x -1 - sqrt{3})

Determining the Domain

The roots of the quadratic equation divide the number line into three intervals:

((-infty, -1 - sqrt{3})) ((-1 - sqrt{3}, -1 sqrt{3})) ((-1 sqrt{3}, infty))

To find the domain, we need to check which intervals make the expression under the square root non-negative. By testing points within these intervals, we find that:

For (x) in ((-1 - sqrt{3}, -1 sqrt{3})), the expression

(2-2x-x^2)

is non-negative. Therefore, the domain of the expression is:

([-1 - sqrt{3}, -1 sqrt{3}])

This means all x values within this interval will result in a real and defined output for the function.

Summary and Conclusion

By solving the given quadratic expression and ensuring the expression under the square root is non-negative, we have determined the domain of (sqrt{2-2x-x^2}) to be:

([-1 - sqrt{3}, -1 sqrt{3}])

This interval represents the complete set of allowable x-values for which the function is real and defined.