Exploring the Domain and Range of Inverse Functions

The video tutorial explains inverse functions, focusing on how two functions can undo each other. It discusses the domain and range of functions, highlighting the need for the domain of one function to match the range of its inverse. Graphical representations are used to illustrate these concepts, and the video demonstrates how restricting the domain can help establish an inverse relationship between functions.

What is the main topic discussed in the video?

• Linear equations

• Inverse functions

• Trigonometric identities

• Quadratic functions

Which two functions are being considered in the video?

• f(x) = e^x and g(x) = ln(x)

• f(x) = sin(x) and g(x) = cos(x)

• f(x) = x^3 and g(x) = x^2

• f(x) = x^2 and g(x) = sqrt(x)

What is the domain of f(x) = x^2?

• Zero to infinity

• Negative infinity to infinity

• Negative infinity to zero

• Zero to one

What is the range of g(x) = sqrt(x)?

• Negative infinity to zero

• Zero to infinity

• Zero to one

• Negative infinity to infinity

What test is used to determine if a function is one-to-one?

• Horizontal line test

• Slope test

• Derivative test

• Vertical line test

What does the horizontal line test check for?

• If a horizontal line intersects the graph more than once

• If a vertical line intersects the graph more than once

• If the function is increasing

• If the function is decreasing

What relationship should the graphs of inverse functions have?

• Reflections over the line y = x

• Parallel lines

• Perpendicular lines

• Identical graphs

How can we make the functions f(x) = x^2 and g(x) = sqrt(x) inverses of each other?

• Restrict the domain of f(x) to [0, infinity)

• Extend the domain of f(x) to (-infinity, infinity)

• Extend the range of g(x) to (-infinity, infinity)

• Restrict the range of g(x) to [0, 1]

What is the restricted domain of f(x) = x^2 to make it an inverse of g(x) = sqrt(x)?

• Negative infinity to infinity

• Negative infinity to zero

• Zero to infinity

• Zero to one

What must be true for two functions to be inverses of each other?

• They must have the same range

• They must be linear

• They must have the same domain

• The domain of one must equal the range of the other