Inverse Functions and Domain Restrictions

This video tutorial explains how to restrict the domain of functions to ensure their inverses are also functions. It covers two examples: a parabola and an absolute value function. The tutorial demonstrates how to apply the vertical and horizontal line tests, interchange x and y, and solve for the new y to find the inverse. It also explains how domain and range switch between the original and inverse functions.

What is the purpose of restricting the domain of a function?

• To make the function continuous

• To ensure the inverse is a function

• To simplify the function

• To make the function differentiable

In the first example, what shape does the graph of f(x) = (x + 3)^2 take?

• A straight line

• A circle

• A parabola

• An ellipse

What does the horizontal line test determine?

• If a graph is differentiable

• If the inverse of a graph is a function

• If a graph is continuous

• If a graph is a function

How can the domain of f(x) = (x + 3)^2 be restricted to make its inverse a function?

• By considering only the right branch of the graph

• By considering only the left branch of the graph

• By considering both branches of the graph

• By considering neither branch of the graph

What happens to the domain and range when finding the inverse of a function?

• They both become zero

• They remain the same

• They both become infinite

• They switch places

In the second example, what transformation is applied to the graph of f(x) = 2|x - 1| + 2?

• Shifted right and up

• Shifted left and down

• Shifted right and down

• Shifted left and up

How is the domain of the absolute value function restricted to make its inverse a function?

• By considering both branches

• By considering neither branch

• By considering only the right branch

• By considering only the left branch

What is the equation of the inverse function for the right branch of f(x) = 2|x - 1| + 2?

• y = 2x

• y = -2x + 4

• y = -1/2x + 2

• y = 1/2x

What is the significance of the line y = x in finding the inverse of a function?

• It is the axis of symmetry

• It is the line of reflection

• It is the line of translation

• It is the line of rotation

What is the domain of the inverse function for the left branch of f(x) = 2|x - 1| + 2?

• x ≤ 2

• x ≥ 1

• x ≤ 1

• x ≥ 2