Question

Define onto function and give an example

Answer 1

An **onto function** (also known as a surjective function) is a type of function in which every element of the codomain has at least one pre-image in the domain. In other words, for a function \( f: A \to B \) to be onto, for every element \( b \) in the set \( B \), there exists at least one element \( a \) in the set \( A \) such that \( f(a) = b \).

### Example of an Onto Function

Consider the function \( f: \mathbb{R} \to \mathbb{R} \) defined by:

\[
f(x) = 2x + 3
\]

To check if this function is onto, we need to see if for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R} \) such that \( f(x) = y \).

1. Start with the equation \( y = 2x + 3 \).
2. Solve for \( x \):

\[
y - 3 = 2x
\]
\[
x = \frac{y - 3}{2}
\]

Since \( x \) can take any real number value for any real number \( y \), it follows that for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R} \) such that \( f(x) = y \). Therefore, the function \( f(x) = 2x + 3 \) is onto.

### Visualizing

Graphically, the line represented by the equation \( y = 2x + 3 \) will cover all real values of \( y \) as \( x \) varies across all real numbers, confirming that every \( y \) in the codomain is achieved. Thus, this function is an example of an onto function.