Inverse Functions and Their Properties

The video tutorial explains how to find the inverse of a given function, including determining the domain and range of both the function and its inverse. It begins by discussing the domain and range of the original function, using interval notation to express them. The tutorial then walks through the process of finding the inverse function by interchanging variables and solving for y. It emphasizes the importance of understanding the domain and range of the original function to correctly determine those of the inverse. Finally, the video demonstrates how to verify the inverse function graphically by plotting both the function and its inverse, showing their reflection across the line y = x.

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

• Multiply the function by a constant

• Add a constant to the function

• Replace the function with y

• Differentiate the function

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

• (-∞, 0)

• (0, ∞)

• (-∞, ∞)

• [0, ∞)

What operation is performed after interchanging x and y to solve for y in the inverse function?

• Multiply both sides by 4

• Subtract 4 from both sides

• Divide both sides by 4

• Add 4 to both sides

What is the final step in expressing the inverse function?

• Replace x with g inverse of y

• Replace x with f inverse of y

• Replace y with g inverse of x

• Replace y with f inverse of x

What is the domain of the inverse function g inverse of x?

• The same as the range of f of x

• The same as the domain of g of x

• The same as the range of g of x

• The same as the domain of f of x

What is the range of the inverse function g inverse of x?

• The same as the range of f of x

• The same as the domain of f of x

• The same as the range of g of x

• The same as the domain of g of x

What is the significance of graphing a function and its inverse on the same coordinate plane?

• They should be parallel lines

• They should intersect at the origin

• They should be reflections across y = x

• They should have the same slope

In the graphical representation, what color is used for the line y = x?

• Purple

• Green

• Blue

• Red

What is the purpose of using a rational exponent instead of a cube root in the graph?

• To change the domain

• To make the graph more complex

• To simplify the graph

• To change the range

What does the reflection of the function and its inverse across y = x verify?

• The slope of the function

• The correctness of the inverse function

• The range of the function

• The domain of the function