Injective, surjective and bijective functions are fundamental concepts in mathematics and are frequently addressed in ENEM questions. To understand these functions, it's important to first understand what a function is.
A function is a relation between two sets, in which each element of the first set (domain) corresponds to a single element of the second set (codomain). That said, let's understand each of the functions.
Injector Functions
A function f is said to be injective (or injective) if and only if different domain elements have different ranges in the range. In other words, no two different values in the domain correspond to the same value in the range. Mathematically, this is expressed as: if x1 ≠ x2, then f(x1) ≠ f(x2).
For example, the function f(x) = 2x + 3 is a one-to-one function. If you take two different values of x, say 1 and 2, you will get two different values of f(x), which are 5 and 7 respectively.
Surjective Functions
A function f is said to be surjective (or surjective) if and only if every element in the range is an image of at least one element in the domain. In other words, there are no values in the range that don't match some value in the domain. Mathematically, this is expressed as: for every y in the range, there is an x in the domain such that f(x) = y.
For example, the function f(x) = x² is a surjective function if we consider the domain and range as the sets of all real numbers. Every real number is the square of some other real number.
Bijection Functions
A function f is said to be one-to-one (or one-to-one) if and only if it is both one-to-one and one-to-one. In other words, each domain element corresponds to a single co-domain element, and vice versa. Mathematically, this is expressed as: if x1 ≠ x2, then f(x1) ≠ f(x2) and for every y in the range, there is an x in the domain such that f(x) = y.
For example, the function f(x) = 2x + 3 is a one-to-one function if we consider the domain and range as the sets of all real numbers. Every real number is the result of 2x + 3 for some real number x, and no two different real numbers result in the same value of f(x).
Understanding one-to-one, one-to-one, and one-to-one functions is crucial to solving many math problems. These concepts are fundamental to the study of functions, which is an important part of the ENEM mathematics curriculum. I hope this text has helped clarify these concepts.
