Inverse Functions and Their Properties

The video tutorial explains how to find the inverse of a given function, f(x) = (x + 2)^2 + 1, with a domain constraint of x ≥ -2. It discusses the importance of domain constraints for finding inverses and demonstrates the step-by-step process of solving for x in terms of y. The tutorial also covers the domain and range considerations for the inverse function and provides a graphical representation, highlighting the mirror image property around the line y = x.

Why is the domain of the function f(x) = (x + 2)^2 + 1 constrained to x >= -2?

• To simplify the calculation of the inverse

• To avoid negative values in the range

• To make the function continuous

• To ensure the function is one-to-one

What is the first step in finding the inverse of the function f(x) = (x + 2)^2 + 1?

• Divide both sides by 2

• Multiply both sides by 2

• Subtract 1 from both sides

• Add 1 to both sides

When solving for x in terms of y, why is it important to consider the positive square root?

• Because x is always positive

• Because the function is decreasing

• Because x + 2 is always non-negative

• Because y is always positive

What is the range of y-values for the function f(x) = (x + 2)^2 + 1?

• y <= 0

• y <= 1

• y >= 1

• y >= 0

How is the inverse function expressed in terms of y?

• x = sqrt(y - 1) + 2

• x = (y - 1)^2 + 2

• x = (y - 1)^2 - 2

• x = sqrt(y - 1) - 2

What is the inverse function of f(x) = (x + 2)^2 + 1 expressed in terms of x?

• f^(-1)(x) = (x - 1)^2 - 2

• f^(-1)(x) = (x - 1)^2 + 2

• f^(-1)(x) = sqrt(x - 1) - 2

• f^(-1)(x) = sqrt(x - 1) + 2

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

• x >= 1

• x >= 0

• x <= 1

• x <= 0

What is the first point plotted on the graph of the inverse function?

• (0, 1)

• (1, -2)

• (0, -1)

• (1, 0)

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

• It is a reflection across the x-axis

• It is identical to the original function

• It is a reflection across the y-axis

• It is a mirror image across the line y = x

What is the significance of the line y = x in relation to the inverse function?

• It is the axis of symmetry for the inverse function

• It is the line where the inverse function is undefined

• It is the line where the original function is undefined

• It is the axis of symmetry for the original function