Understanding Inverses of Functions

Mr. Allen introduces inverse functions, focusing on quadratic and cubic functions. He explains how to find inverses, determine if they are functions, and analyze their domain, range, and intercepts. The video covers graphing techniques and the importance of domain restrictions to ensure inverses are functions. The reflection over the line y=x is highlighted, and the video concludes with a preview of future topics like transformations.

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

• Plotting the function on a graph

• Swapping the x and y variables

• Finding the domain and range

• Applying the vertical line test

When finding the inverse of a quadratic function, why do we include a plus or minus sign?

• To pass the vertical line test

• To ensure the function is invertible

• To simplify the equation

• To account for both positive and negative roots

How can you determine if an inverse is a function using a graph?

• By confirming it has a domain of all real numbers

• By verifying it passes the vertical line test

• By ensuring it is symmetric over the y-axis

• By checking if it passes the horizontal line test

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

• They remain unchanged

• They are swapped

• They are both set to all real numbers

• They are restricted to positive values

How can restricting the domain of a quadratic function affect its inverse?

• It can change the range of the inverse

• It can make the inverse symmetric over the y-axis

• It can make the inverse a function

• It can eliminate the need for a plus or minus sign

What is the domain of the inverse of a quadratic function?

• All real numbers

• x is greater than or equal to zero

• x is less than or equal to zero

• x is equal to zero

What is a key difference between the inverses of quadratic and cubic functions?

• Quadratic inverses have a domain of all real numbers

• Quadratic inverses are always functions

• Cubic inverses do not require a plus or minus sign

• Cubic inverses are not symmetric over y = x

Why is the inverse of a cubic function always a function?

• Because it does not require a plus or minus sign

• Because it has a domain of all real numbers

• Because it is symmetric over the y-axis

• Because it passes the vertical line test

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

• It is the axis of symmetry for all functions

• It is the line over which inverses reflect

• It is the line that determines the domain

• It is the line that all functions must pass through

What is the range of the inverse of a cubic function?

• All real numbers

• y is greater than or equal to zero

• y is less than or equal to zero

• y is equal to zero