the graph of the function alone; that is, whether a function is injective can be decided by only considering the graph (and not the codomain) of Proving that functions are injective A proof that a function is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if then Here is an example: Proof: Let Suppose So implies which implies Therefore, it follows from the definition that is injective. There are multiple other methods of proving that a function is injective. For example, in calculus if is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if is a linear transformation it is sufficient to show that the kernel of contains only the zero vector. If is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list. A graphical approach for a real-valued function of a real variable is the horizontal line test. If every horizontal line intersects the curve of in at most one point, then is injective or one-to-one. Gallery See also Notes References , p. 17 ff. , p. 38 ff. External links Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms. Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injective functions Functions and mappings Basic concepts in set theory Types of functions