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In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f ( x ) = x is true for all values of x to which f can be applied.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
A function : is bijective if and only if it is invertible; that is, there is a function : such that = identity function on and = identity function on . This function maps each image to its unique preimage.
Identity (mathematics), an equality that holds regardless of the values of its variables; Identity element, an element of the set which leaves unchanged every element when the operation is applied; Identity function, a function that leaves its argument unchanged; Identity matrix, with ones on the main diagonal, zeros elsewhere
In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity. To see this, note that if l is a left identity and r is a right identity, then l = l ∗ r = r.
constant functions are idempotent; the identity function is idempotent; the floor, ceiling and fractional part functions are idempotent; the real part function () of a complex number, is idempotent. the subgroup generated function from the power set of a group to itself is idempotent;
In particular, the identity function is always injective (and in fact bijective). If the domain of a function is the empty set, then the function is the empty function, which is injective. If the domain of a function has one element (that is, it is a singleton set), then the function is always injective.