Inverse Functions and Graphs

Front

What is an inverse function?

Back

An inverse function reverses the operation of the original function. If f(x) = y, then the inverse function f⁻¹(y) = x.

Front

How can you determine if a function has an inverse?

Back

A function has an inverse if it passes the horizontal line test, meaning no horizontal line intersects the graph more than once.

Front

What is the horizontal line test?

Back

The horizontal line test is a method to determine if a function is one-to-one. If any horizontal line crosses the graph more than once, the function does not have an inverse.

Front

What does it mean for two functions to be inverses?

Back

Two functions are inverses if applying one function followed by the other returns the original input. For example, f(f⁻¹(x)) = x.

Front

What is the graphical representation of an inverse function?

Back

The graph of an inverse function is a reflection of the original function's graph over the line y = x.

Front

What is the line of reflection for inverse functions?

Back

The line of reflection for inverse functions is y = x.

Front

Can a function be its own inverse?

Back

Yes, a function can be its own inverse if f(f(x)) = x for all x in the domain.

Front

What is the relationship between the domains and ranges of a function and its inverse?

Back

The domain of the original function becomes the range of the inverse function, and vice versa.

Front

What is a one-to-one function?

Back

A one-to-one function is a function where each output is produced by exactly one input, ensuring it has an inverse.

Front

What is the significance of the inverse of a function in real-world applications?

Back

Inverse functions are used in various applications, such as calculating the original value from a transformed value in finance, science, and engineering.