For each function, a) determine whether it is one-to-one; b)...

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Key Concepts

Transformations of Functions

Understanding how basic functions are transformed is important when dealing with inverses. For example, when a function undergoes a horizontal shift, vertical shift, or other modifications, these changes affect its graph as well as the process of finding its inverse. The concept is crucial in recognizing that if the transformation applied is reversible, the inverse function can be determined by applying the corresponding reverse transformation.

Inverse Function

An inverse function essentially reverses the operation of the original function. If f is a one-to-one function, then its inverse f?¹ will satisfy the condition that f(f?¹(x)) = x and f?¹(f(x)) = x for all x in the appropriate domains. Finding the inverse usually involves interchanging the roles of x and y and solving for y.

One-to-One Function

A one-to-one function, also known as an injective function, is one in which every element of the function's domain maps to a unique element in the codomain. In other words, no two different inputs produce the same output. This property is critical when determining if a function can have an inverse, because only one-to-one functions have well-defined inverses.

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