Find the domain of the function f(x)=(1)/(sqrt(2-2x-x^(2))). (Enter your answer using interval notation.)

Identify Inner Function: Identify the inner function within the square root to determine where it is non-negative, as the square root function requires non-negative inputs for real number outputs. Rewrite in Standard Form: Rewrite the quadratic equation in standard form and identify values of a, b, and c. Check Discriminant: Use the discriminant formula, b^2 - 4ac, to check if the quadratic equation has real roots, which will help in finding the intervals where the function under the square root is non-negative. Solve for Roots: Since the discriminant is positive, solve for the roots of the equation using the quadratic formula, x = (-b ± sqrt(b^2 - 4ac))/(2a). Determine Non-Negative Intervals: Determine the intervals where the function under the square root, 2 - 2x - x^2, is non-negative. This is between the roots because the parabola opens downwards (a = 1). Write Domain: Write the domain of f(x) using interval notation, considering where the function under the square root is non-negative.

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Find the domain of the function
f(x)=(1)/(sqrt(2-2x-x^(2))). (Enter your answer using interval notation.)

Find the domain of the function $f(x)=\frac{1}{\sqrt{2-2 x-x^{2}}}$ . (Enter your answer using interval notation.)

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Algebra 1

Domain and range of quadratic functions: equations

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Q. Find the domain of the function

f(x)=\frac{1}{\sqrt{2-2 x-x^{2}}}

. (Enter your answer using interval notation.)

Identify Inner Function: Identify the inner function within the square root to determine where it is non-negative, as the square root function requires non-negative inputs for real number outputs.
Rewrite in Standard Form: Rewrite the quadratic equation in standard form and identify values of $a$ , $b$ , and $c$ .
Check Discriminant: Use the discriminant formula, $b^2 - 4ac$ , to check if the quadratic equation has real roots, which will help in finding the intervals where the function under the square root is non-negative.
Solve for Roots: Since the discriminant is positive, solve for the roots of the equation using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Determine Non-Negative Intervals: Determine the intervals where the function under the square root, $2 - 2x - x^2$ , is non-negative. This is between the roots because the parabola opens downwards ( $a = -1$ ).
Write Domain: Write the domain of $f(x)$ using interval notation, considering where the function under the square root is non-negative.