Domain and Inverse Functions

The video tutorial explains how to determine the inverse of a function when domain restrictions are applied. It covers graphing techniques for quadratics with restricted domains, the algebraic process of finding inverses, and the importance of transformations. The tutorial also provides examples to illustrate how domain restrictions affect the inverse function, ensuring it remains a function.

What is the unrestricted domain of the function f(x) = 3x^2 - 2?

• Negative infinity to 0

• Negative infinity to infinity

• 0 to infinity

• 0 to 10

What is the domain restriction applied to the function f(x) = 3x^2 - 2?

• x <= 0

• x < 0

• x >= 0

• x > 0

Why is it important to consider only the positive values of the function f(x) = 3x^2 - 2?

• To simplify the graph

• To ensure the function remains a function

• To make calculations easier

• To avoid negative values

What is the first step in finding the inverse of the function f(x) = 3x^2 - 2?

• Swap x and y

• Add 2 to both sides

• Divide by 3

• Take the square root

Why do we include a plus or minus sign when taking the square root in the process of finding the inverse?

• To simplify the equation

• To avoid complex numbers

• To account for both positive and negative roots

• To make the function continuous

What transformation does the term 1/3 in the inverse function indicate?

• Vertical stretch

• Horizontal stretch

• Horizontal compression

• Vertical compression

What happens if the domain restriction is not applied to the function f(x) = 3x^2 - 2?

• The function becomes a line

• The function remains the same

• The function becomes undefined

• The function becomes a sideways parabola

In the second example, what is the domain restriction for the quadratic function in vertex form?

• x >= -2

• x < -2

• x <= -2

• x > -2

What is the final inverse function for the second example?

• f^-1(x) = sqrt(x - 1) - 2

• f^-1(x) = -sqrt(x - 1) - 2

• f^-1(x) = sqrt(x + 1) - 2

• f^-1(x) = -sqrt(x + 1) - 2

How can you verify the correctness of the inverse function using Desmos?

• By checking if the graph is symmetrical about y = 0

• By checking if the graph is symmetrical about y = x

• By checking if the graph is a parabola

• By checking if the graph is a straight line