One-to-one function

Definition

A one-to-one function, also known as an injective function, is a type of function where each element in the domain maps to a unique element in the codomain, ensuring that no two different inputs produce the same output. This property is crucial when discussing composition and inverse functions, as it guarantees that the inverse function exists and is itself a valid function. Understanding one-to-one functions helps in determining how functions interact with one another and allows for the clear identification of relationships between inputs and outputs.

5 Must Know Facts For Your Next Test

1. For a function to be one-to-one, it must pass the horizontal line test; no horizontal line should intersect the graph of the function more than once.
2. If a function is one-to-one, then its inverse will also be a function, meaning each output corresponds to exactly one input.
3. Not all functions are one-to-one; many functions like quadratics or polynomials can map multiple inputs to the same output.
4. One-to-one functions can be represented algebraically; if \( f(a) = f(b) \) implies \( a = b \), then \( f \) is one-to-one.
5. The composition of two one-to-one functions is also one-to-one, maintaining the uniqueness of outputs from the combined inputs.

Review Questions

• How does understanding whether a function is one-to-one impact the ability to find its inverse?
  • Knowing that a function is one-to-one is essential for finding its inverse because only injective functions guarantee that each output corresponds to a single input. This means that if you can determine that a function is one-to-one, you can confidently derive its inverse without ambiguity or duplication in outputs. If a function fails to be one-to-one, it will not have a proper inverse, as some outputs would correspond to multiple inputs.
• Discuss how the horizontal line test relates to identifying one-to-one functions and give an example of a function that fails this test.
  • The horizontal line test is a visual way to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, then the function cannot be classified as one-to-one. A classic example of this is the quadratic function \( f(x) = x^2 \); any horizontal line above the x-axis will intersect this parabola at two points, indicating it is not injective.
• Evaluate how composition affects one-to-one functions and provide an example illustrating this concept.
  • When composing functions, if both functions involved are one-to-one, their composition will also be one-to-one. This principle ensures that the uniqueness of each output is preserved throughout the combined operation. For instance, if \( f(x) = 2x + 3 \) and \( g(x) = x - 5 \) are both one-to-one functions, then their composition \( (f \, ext{circ} \, g)(x) = f(g(x)) = 2(x - 5) + 3 = 2x - 7 \) remains injective as well.

Related terms

Injective Function: A synonym for one-to-one function, emphasizing that distinct inputs yield distinct outputs.

Inverse Function: A function that reverses the effect of another function, existing only if the original function is one-to-one.

Surjective Function: A function that covers every element in the codomain at least once, contrasting with one-to-one functions which focus on unique mappings.