Math Lesson 16.2.2 - Injective Function

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Welcome to our Math lesson on Injective Function, this is the second lesson of our suite of math lessons covering the topic of Injective, Surjective and Bijective Functions. Graphs of Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Injective Function

Based on the relationship between variables, functions are classified into three main categories (types).

The first type of function is called injective; it is a kind of function in which each element of the input set X is related to a distinct element of the output set Y. It is a kind of one-to-one function, but where not all elements of the output set are connected to those of the input set. The following diagram shows an example of an injective function where numbers replace numbers

As you see, all elements of input set X are connected to a single element from output set Y. However, the output set contains one or more elements not related to any element from input set X. Therefore, this is an injective function.

As an example of the injective function, we can state f(x) = 5 - x {x ∊ N, Y ∊ N, x ≤ 4, y ≤ 5} is an injective function because all elements of input set X have, in correspondence, a single element of the output set Y. However, one of the elements of the set Y (y = 5) is not related to any input value because if we write 5 = 5 - x, we must have x = 0. This is a value that does not belong to the input set. The following figure shows this function using the Venn diagram method.

An injective function cannot have two inputs for the same output. For example, f(x) = x^x is not an injective function in Z because for x = -5 and x = 5 we have the same output y = 25.

The formal definition of injective function is as follows:

"A function f is injective only if for any f(x) = f(y) there is x = y."

The quadratic function above does not meet this requirement because for x = -5 ≠ x = 5 but both give f(x) = f(y) = 25.

Example 2

Which of the following functions is injective? If not, prove it through a counter-example.

1. f(x) = 3x - 2 (x ∊ Z, y ∊ Z)
2. g(x) = 5/(2x + 1) (x ∊ Z, y ∊ Z)
3. h(x) = x4 - 1 (x ∊ R, y ∊ R)

Solution 2

1. This function is injective because for every x there is a different y. This is obvious because the function is linear, i.e. f(x) = f(y) if only x = y. For example, if x = 5 and y = 5, then
   f(x) = f(5)
   = 3 ∙ 5 - 2
   = 15 - 2
   = 13
   and
   f(y) = f(5)
   = 3 ∙ 5 - 2
   = 15 - 2
   = 13
2. This is not an injective function, as, for example, for x = 1, then
   f(1) = 5/2 ∙ 1 + 1
   = 5/2 + 1
   = 5/3
   This result (output) is not an integer, so this function is not injective because we found an element of the input set X which has not in correspondence an element of the output set Y.
3. This is not an injective function because we can find two different elements of the input set X which have in correspondence a single element of the output set Y. For example, for the inputs x = 2 and x = -2 the result (output) is f(-2) = f(2) because f(-2) = (-2)⁴ - 1 = 16 - 1 = 15 and f(2) = 24 - 1 = 16 - 1 = 15.

You have reached the end of Math lesson 16.2.2 Injective Function. There are 7 lessons in this physics tutorial covering Injective, Surjective and Bijective Functions. Graphs of Functions, you can access all the lessons from this tutorial below.

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