Onto Function (Surjection) -Definition and Examples

Here you will learn what is onto function with definition and examples.

Let’s begin –

Onto Function (Surjection) Definition

Definition : A function f : A \(\rightarrow\) B is said to be an onto function if every element of B is the f-image of some element of A i.e. , if f(A) = B or range of f is the codomain of f.

Thus, f : A \(\rightarrow\) B is a surjection iff for each b \(\in\) B, there exist a \(\in\) A such that f(a) = b

Also Read : Types of Functions in Maths – Domain and Range

Example : Let g : X \(\rightarrow\) Y be the function represented by the following diagram :

onto function

Solution : Under function g every element in Y has its pre-image X. So, g : X \(\rightarrow\) Y is onto.

Algorithm to Check for Onto

Let f : A \(\rightarrow\) B be the given function.

1).  Choose an arbitrary element y in b.

2).  Put f(x) = y

3).  Solve the equation f(x) = y for x and obtain x in terms of y. Let x = g(y).

4).  If for all values of y \(\in\) B, the values of x obtained from x = g(y) are in A, then f is onto.

Note : If range is same as codomain, then f is onto function.

Example : Let f : R \(\rightarrow\) R given by f(x) = \(x^3 + 2\) for all x \(\in\) R. Then, find whether it is onto or not.

Solution : Let y be the arbitrary element of R. Then,

f(x) = y \(\implies\) \(x^3 + 2\) = y \(\implies\) x = \((y – 2)^{1/3})\)

Clearly, for all y \(\in\) R, \((y – 2)^{1/3})\) is a real number. Thus, for all y \(\in\) R (co-domain) there exist x = \((y – 2)^{1/3})\) in R (domain) such that f(x) = \(x^3 + 2\) = y.

Hence, f : R \(\rightarrow\) R is an onto function.

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