Understanding Functions and Their Inverses

The video tutorial explains how to determine the domain and range of a function involving a square root, and how to find its inverse. It covers the restrictions necessary for the function to be real, solving inequalities to find the domain, and using interval notation. The tutorial also explains how to find the range by considering possible y-values and verifies these graphically. The process of finding the inverse function involves interchanging variables and solving for y, with a focus on understanding the relationship between the domain and range of the original and inverse functions. Graphical verification is used to confirm the symmetry of the function and its inverse across the line y=x.

What is the function given in the video?

• f(x) = √(2x + 1) - 3

• f(x) = √(2x - 1) - 3

• f(x) = √(2x - 1) + 3

• f(x) = √(2x + 1) + 3

What condition must be met for the radicand in the function to ensure real values?

• 2x - 1 must be less than zero

• 2x - 1 must be greater than zero

• 2x - 1 must be greater than or equal to zero

• 2x - 1 must be equal to zero

What is the domain of the function f(x) = √(2x - 1) - 3?

• x ≥ 1

• x ≥ 0

• x ≥ -1/2

• x ≥ 1/2

How is the domain of the function expressed in interval notation?

• (-∞, 1/2]

• [1/2, ∞)

• (1/2, ∞)

• [0, ∞)

What is the range of the function f(x) = √(2x - 1) - 3?

• y ≥ -1

• y ≥ 3

• y ≥ -3

• y ≥ 0

How is the range of the function expressed in interval notation?

• (-∞, -3]

• (-3, ∞)

• [-3, ∞)

• [0, ∞)

What is the first step in finding the inverse of the function?

• Add 3 to both sides

• Subtract 3 from both sides

• Multiply both sides by 2

• Divide both sides by 2

What is the inverse function of f(x) = √(2x - 1) - 3?

• f⁻¹(x) = 2(x + 3)² + 1

• f⁻¹(x) = 1/2(x + 3)² + 1

• f⁻¹(x) = 1/2(x + 3)² - 1

• f⁻¹(x) = 1/2(x - 3)² + 1

What is the domain of the inverse function?

• x ≥ 1/2

• x ≥ -3

• x ≥ 0

• x ≥ 3

What line should the graphs of the function and its inverse be symmetrical across?

• x = y

• y = x

• x = 0

• y = 0