Onto Function 

In mathematics, an onto function, also known as a surjective function, is a type of mapping where every element of the codomain has at least one corresponding element in the domain. This means the range of the function is equal to its codomain. Onto functions are essential in understanding advanced concepts like inverse functions and bijections. They are widely used in algebra, calculus, and real-world applications involving full data coverage or target mapping.

1.0Onto Function Definition

In mathematics, an onto function (also called a surjective function) is a type of function where every element of the codomain is mapped by at least one element of the domain.

In simple terms, a function $f:A\to B$ is onto if range(f) = codomain(B).

2.0What Is the Onto Function?

An onto function ensures that no element in the codomain is left unmapped. This makes it a surjective function. In set theory, it means for every $b\in B$, there exists at least one a $\in$ A such that f(a) = b.

3.0Onto Function Formula

Let f: $A\to B$. Then f is onto if:

$\forall b\in B$, $\exists a\in A$ such that f(a) = b 

This is the onto function condition for surjectivity.

4.0How Do You Know If a Function Is Onto?

To check if a function is onto:

1. Solve the equation f(x) = y for x
2. Verify whether a solution exists for every y in the codomain
3. If every value in the codomain has a pre-image in the domain, the function is onto

5.0Onto Function Graph

In a graph of an onto function, every y-value in the codomain is covered by the graph. For real-valued functions, this means the range spans the entire codomain.

For example, the function:

$f(x)=2x+3,\text{ where }f:⊠\to ⊠$

is onto because for every real number y, we can find an $x\in \mathbb{X}$ such that:

$x=\frac{y-3}{2}$ 

Hence, every y has a corresponding x: the function is onto.

6.0Onto Function Properties

• Every onto function is also called a surjection
• The codomain = range
• Inverse functions exist only if the function is one-one and onto
• Composition of two onto functions is also onto
• Not all functions are onto; it depends on how domain and codomain are defined

7.0Solved Examples on Onto Function 

Example 1: Is the function f(x) = 2x + 3, defined as $f:⊠\to ⊠$, an onto function?

Solution:

To check if f is onto, solve for x in terms of y:

$y=2x+3\Rightarrow x=\frac{y-3}{2}$

Since for every $y\in ⊠$, there exists an $x\in ⊠$, the function is onto.

Answer: The function is onto.

Example 2: Check if $f(x)=x^2$, defined as $f:⊠\to ⊠$, is an onto function.

Solution:
The range of $f(x)=x^{2}is[0,\infty )$, not the entire codomain $⊠$ .

Negative values like y = -1 have no real pre-image.

Answer: The function is not onto.

Example 3: Is $f(x)=e^x$, defined as $f:⊠\to ⊠$, onto?

Solution:

The range of $f(x)=e^{x}is(0,\infty )$, so values like y = -1 are not covered.

Answer: The function is not onto.

Example 4: Let A = {1, 2, 3}, B = {a, b}, and $f:A\to B$ defined by Is f onto?

Solution:

Elements of B: a and b are both mapped from elements in A. Hence, the codomain is completely covered.

Answer: The function is onto.

Example 5: Check if $f(x)=\frac{x-1}{x+1}$, defined as $f:⊠,\{-1\}\to ⊠,\{1\}$, is onto.

Solution:
Let

$y=\frac{x-1}{x+1}$

Solve for x:

$\begin{array}{rl}& y(x+1)=x-1\\ & \Rightarrow yx+y=x-1\\ & \Rightarrow yx-x=-1-y\\ & \Rightarrow x(y-1)=-1-y\\ & \Rightarrow x=\frac{-1-y}{y-1}\end{array}$

The expression is valid for all $y\ne 1$, which matches the codomain.

Answer: The function is onto.

8.0Practice Questions on Onto Functions

1. Check if $f(x)=x^2+1,f:⊠\to ⊠,$ is onto.
2. Is $f(x)=\ln (x),f:(0,\infty )\to ⊠,$ an onto function?
3. Define a mapping from set A = {1, 2, 3} to B = {a, b} that is onto.
4. Prove $f(x)=\frac{x-1}{x+1},f:⊠,\{-1\}\to ⊠,$, is onto.
5. Differentiate between into and onto with proper sets and mappings.

9.0One-One and Onto Function (Bijective Function)

A function that is both one-one (injective) and onto (surjective) is called bijective.

Properties:

• Each element of the codomain is mapped exactly once
• A bijective function has an inverse
• Useful in establishing one-to-one correspondence between sets

10.0Differences Between Onto and Into Functions