One-to-One Function

Definition

A one-to-one function, also known as an injective function, is a special type of function where each element in the domain is paired with a unique element in the codomain. In other words, for any two distinct elements in the domain, their corresponding elements in the codomain must also be distinct.

5 Must Know Facts For Your Next Test

1. A one-to-one function has the property that for any two distinct elements in the domain, their corresponding elements in the codomain are also distinct.
2. One-to-one functions are important in the context of inverse functions, as they guarantee a unique inverse function can be defined.
3. Transformations of functions, such as reflections, can preserve the one-to-one property of a function.
4. Radical functions, such as $\sqrt{x}$, are not one-to-one functions, but their inverse functions, such as $x^2$, are one-to-one.
5. Testing for one-to-one functions can be done using the horizontal line test, where a function is one-to-one if no horizontal line intersects the graph more than once.

Review Questions

• Explain how the one-to-one property of a function is related to the concept of function notation and the domain and codomain.
  • The one-to-one property of a function is closely tied to the concept of function notation and the domain and codomain of a function. For a function to be one-to-one, each element in the domain must be paired with a unique element in the codomain. This means that the function notation $f(x)$ will always return a single, unique output value for any given input value $x$ in the domain. The one-to-one property ensures that the function notation is well-defined and that there is a clear, unambiguous relationship between the input and output values.
• Describe how the one-to-one property of a function is related to the transformation of functions, particularly reflections.
  • The one-to-one property of a function is an important consideration when transforming functions, such as through reflections. Reflections, which involve flipping a function across a line, can preserve the one-to-one property of a function. For example, reflecting the function $f(x) = x^2$ across the line $y = x$ results in the inverse function $g(x) = \sqrt{x}$, which is also a one-to-one function. This relationship between one-to-one functions and their inverses is crucial in understanding how transformations of functions can impact their properties and behavior.
• Explain the connection between one-to-one functions and inverse functions, particularly in the context of radical functions.
  • The one-to-one property of a function is essential for the existence of an inverse function. If a function is one-to-one, then it can be guaranteed that a unique inverse function can be defined, which maps the output values back to the original input values. This relationship is particularly important in the context of radical functions, such as $\sqrt{x}$. While the radical function itself is not one-to-one, its inverse function, $x^2$, is one-to-one. This allows for the inverse function to be well-defined and to provide a unique solution for the original input value. Understanding the connection between one-to-one functions and inverse functions is crucial for working with a variety of mathematical functions, including those involving radicals.

Related terms

Domain: The set of input values for a function, or the set of independent variables.

Codomain: The set of all possible output values for a function, or the set of dependent variables.

Inverse Function: A function that undoes the original function, mapping the output values back to the original input values.