Inverse Functions and Their Properties

The video tutorial explains how to find the inverse of a constrained function, f(x) = (x-1)^2 - 2, with a domain of x ≤ 1. It discusses the importance of understanding domain and range constraints when finding inverses. The tutorial highlights the need to take the negative square root due to the domain constraint, ensuring the correct inverse function is derived. The process involves solving for x in terms of y, swapping x and y, and graphing the inverse function, which reflects the original function across the line y = x.

What is the function f(x) given in the video?

• f(x) = (x + 1)^2 + 2

• f(x) = (x - 1)^2 - 2

• f(x) = (x - 1)^2 + 2

• f(x) = (x + 1)^2 - 2

Why is the domain of the function constrained to x ≤ 1?

• To focus on the left half of the parabola

• To ensure the function is linear

• To simplify the calculation of the inverse

• To make the function a full parabola

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

• Set x equal to y

• Set y equal to x

• Set y equal to f(x)

• Set x equal to f(y)

What is the range of y when solving for the inverse function?

• y ≤ -2

• y ≤ 2

• y ≥ 2

• y ≥ -2

Why is it necessary to take the negative square root when solving for x?

• Because x is greater than or equal to 1

• Because x is less than or equal to 1

• Because x is always negative

• Because x is always positive

What does the negative square root ensure in the context of this problem?

• It ensures the function is linear

• It ensures x is positive

• It ensures the correct inverse function

• It ensures x is negative

What is the inverse function f⁻¹(x) in terms of x?

• f⁻¹(x) = √(x + 2) - 1

• f⁻¹(x) = -√(x + 2) + 1

• f⁻¹(x) = -√(x + 2) - 1

• f⁻¹(x) = √(x + 2) + 1

What is the domain of the inverse function f⁻¹(x)?

• x ≤ -2

• x ≥ -2

• x ≥ 2

• x ≤ 2

How does the graph of the inverse function relate to the original function?

• It is a translation of the original function

• It is a reflection along the line y = x

• It is a reflection along the line y = -x

• It is identical to the original function

What makes this inverse problem particularly challenging?

• The function is linear

• The range constraint is too broad

• The domain constraint requires a negative square root

• The function is quadratic