Question

Explain one -to-one function with an example

Answers

GPT-4o mini
A one-to-one function, also known as an injective function, is a type of function where each element of the codomain is mapped to by at most one element of the domain. This means that if \( f(a) = f(b) \), then it must follow that \( a = b \). In simpler terms, no two different inputs can produce the same output.

### Example

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

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

To check if this function is one-to-one, we can take two arbitrary inputs \( a \) and \( b \) from the domain:

Suppose \( f(a) = f(b) \).

This gives us:

\[
2a + 3 = 2b + 3
\]

Subtracting 3 from both sides:

\[
2a = 2b
\]

Dividing by 2:

\[
a = b
\]

Since we have shown that \( f(a) = f(b) \) implies \( a = b \), it follows that the function \( f(x) = 2x + 3 \) is one-to-one.

### Visual Representation

If you were to graph this function, it would be a straight line with a slope of 2. You would see that any horizontal line drawn through the graph would intersect the line only once, reinforcing that there are no two different inputs that produce the same output. This property is characteristic of one-to-one functions.

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