Inverse Functions and One-to-One Functions

The video tutorial introduces inverse functions, explaining their definitions, properties, and how to determine if a function has an inverse. It emphasizes the importance of one-to-one functions, as only these have inverses. The video provides examples and graphical representations to illustrate the concepts, and concludes with a summary of key properties, including the symmetry of inverse functions across the line y = x.

What is a one-to-one function?

• A function where each y-value is paired with exactly one x-value

• A function where each y-value is paired with multiple x-values

• A function where each x-value is paired with multiple y-values

• A function that fails the vertical line test

Which test helps determine if a function is one-to-one?

• Horizontal Line Test

• Derivative Test

• Slope Test

• Vertical Line Test

Why is it important for a function to be one-to-one?

• Because only one-to-one functions have inverse functions

• Because one-to-one functions are always continuous

• Because one-to-one functions are always linear

• Because one-to-one functions are always quadratic

What is the informal definition of an inverse function?

• A function that squares the input

• A function that multiplies by 2

• A function that undoes another function

• A function that adds 3

If f(x) = 3x, what is the inverse function g(x)?

• g(x) = 3 / x

• g(x) = x + 3

• g(x) = x - 3

• g(x) = x / 3

What is the formal definition of an inverse function?

• A function that maps from set A to set B

• A function that maps from set B to set A and returns each element to itself

• A function that maps from set B to set A

• A function that maps from set A to set B and returns each element to itself

What is the first step in determining an inverse function?

• Graph the function

• Interchange the x and y variables

• Solve the equation for y

• Determine if the function is one-to-one

What is the inverse function of f(x) = x^3 + 2?

• f^-1(x) = (x + 2)^3

• f^-1(x) = (x - 2)^(1/3)

• f^-1(x) = (x + 2)^(1/3)

• f^-1(x) = (x - 2)^3

What is a key graphical property of a function and its inverse?

• They are symmetrical across the line y = x

• They are symmetrical across the line y = 0

• They are symmetrical across the line x = 0

• They are symmetrical across the line y = y

What is the relationship between the domain and range of a function and its inverse?

• The domain of f is the range of f inverse

• The range of f is the domain of f

• The range of f is the range of f inverse

• The domain of f is the domain of f inverse