Understanding the Domain of a Square Root Function

The video tutorial explains how to find the domain of a square root function, specifically f(x) = √(-x² + 4x + 5). It discusses the importance of ensuring the radicand is non-negative for the function to be real. The tutorial demonstrates solving a quadratic inequality to determine the domain, using both algebraic and graphical methods. It involves finding x-values where the expression equals zero, testing intervals, and verifying results graphically. The domain is determined to be the closed interval from -1 to 5.

What is the primary condition for the radicand in a square root function to ensure the function value is real?

• The radicand must be greater than or equal to zero.

• The radicand must be less than zero.

• The radicand must be greater than zero.

• The radicand must be equal to zero.

What is the first step in solving the quadratic inequality -x^2 + 4x + 5 ≥ 0?

• Ignore the inequality and solve directly.

• Test random values for x.

• Determine the x-values where the expression equals zero.

• Graph the function directly.

After factoring the quadratic equation, what are the roots of x^2 - 4x - 5 = 0?

• x = 1 and x = -5

• x = 5 and x = -1

• x = -5 and x = 1

• x = 0 and x = 5

Why do we use closed points at x = -1 and x = 5 on the number line?

• Because the inequality is strict.

• Because these points do not satisfy the inequality.

• Because the inequality includes equal to zero.

• Because these points are not part of the domain.

What is the result of testing the interval with x = 0 for the inequality -x^2 + 4x + 5 ≥ 0?

• The interval is undefined.

• The interval is true.

• The interval is false.

• The interval is not part of the domain.

Which interval is part of the domain after testing x = -2, 0, and 6?

• The interval from -1 to 5

• The interval from 5 to 6

• The interval from 0 to 6

• The interval from -2 to 0

How can the domain of the function be verified graphically?

• By checking when the function is parallel to the x-axis.

• By checking when the function is below the x-axis.

• By checking when the function is equal to the x-axis.

• By checking when the function is above the x-axis.

What does it mean when the function is above the x-axis in terms of the domain?

• The function is negative.

• The function is undefined.

• The function is zero.

• The function is positive.

What is the domain of the function f(x) = √(-x^2 + 4x + 5) using interval notation?

• [-1, 5]

• (-1, 5)

• (-∞, ∞)

• [0, 5]

What is another method to determine the domain of a square root function besides solving algebraically?

• Ignoring the inequality.

• Using a calculator.

• Guessing the values.

• Graphing the original function.