Understanding the Domain of a Function

The video tutorial explains how to determine the domain of the function f(x) = sqrt((x-2)/(x-5)). It covers restrictions from both the numerator and denominator, emphasizing that x cannot equal 5 due to division by zero, and x must be greater than or equal to 2 for the square root to be real. The domain is expressed using inequalities, interval notation, and graphically. The tutorial concludes with a graphical verification of the domain.

What is the primary reason for considering the denominator when determining the domain of a function?

• To ensure the function is continuous.

• To avoid division by zero.

• To simplify the function.

• To find the range of the function.

What value of x must be excluded to avoid division by zero in the function f(x) = √((x-2)/(x-5))?

• x = -5

• x = 0

• x = 2

• x = 5

Why must the expression under the square root in the numerator be non-negative?

• To ensure the result is a real number.

• To ensure the function is continuous.

• To ensure the function is integrable.

• To ensure the function is differentiable.

What is the second restriction on the domain of the function f(x) = √((x-2)/(x-5))?

• x must be greater than or equal to 2.

• x must be less than or equal to 2.

• x must be less than or equal to 5.

• x must be greater than or equal to 5.

Which of the following is a correct way to express the domain of the function using inequalities?

• x > 2 and x ≠ 5

• x ≥ 2 and x ≠ 5

• x ≤ 2 and x ≠ 5

• x < 2 and x ≠ 5

How is the domain of the function represented graphically on a number line?

• An open point at both 2 and 5.

• A closed point at both 2 and 5.

• An open point at 5 and a closed point at 2.

• A closed point at 5 and an open point at 2.

What is the interval notation for the domain of the function f(x) = √((x-2)/(x-5))?

• [2, 5] ∪ (5, ∞)

• (2, 5) ∪ [5, ∞)

• [2, 5) ∪ (5, ∞)

• (2, 5] ∪ [5, ∞)

What is the purpose of projecting the function onto the x-axis?

• To simplify the function.

• To determine the function's continuity.

• To verify the domain is correct.

• To find the range of the function.

What does the graph of the function indicate about the domain between x = 2 and x = 5?

• The function is constant.

• The function is continuous.

• The function is undefined.

• The function is discontinuous.

What does the break at x = 5 on the graph signify?

• A point of continuity.

• A point of discontinuity.

• A point of maximum value.

• A point of minimum value.