One-to-One Function

Definition

A function $f: A \to B$ is called one-to-one or injective if different elements of the domain always map to different elements of the codomain.

Formally, $f$ is one-to-one if:

$$f(a_1) = f(a_2) \Rightarrow a_1 = a_2$$

for all $a_1, a_2 \in A$.

Equivalently,

$$a_1 \ne a_2 \Rightarrow f(a_1) \ne f(a_2)$$

This means no two distinct inputs in the domain share the same output.

Main Content

1. Meaning of Injective Mapping

• A one-to-one function assigns each input a unique output, so there is no repetition in outputs for different inputs.
• If two different elements of the domain produced the same value, then the function would not be one-to-one.

For example, consider $f(x) = 2x + 3$.
If $f(x_1) = f(x_2)$, then:

$$2x_1 + 3 = 2x_2 + 3$$

$$2x_1 = 2x_2$$

$$x_1 = x_2$$

So this function is one-to-one.

Another example is $f(x) = x^2$ on all real numbers. It is not one-to-one because:

$$f(2)=4,\quad f(-2)=4$$

Here, two different inputs give the same output.

2. Ways to Test One-to-One Functions

Algebraic test

• Assume $f(a)=f(b)$ and prove that $a=b$. If this works, the function is injective.

Graphical test

• A function is one-to-one if any horizontal line intersects its graph at most once. This is called the horizontal line test.

Example:

• $f(x)=x^3$ is one-to-one because every horizontal line cuts the graph only once.
• $f(x)=x^2$ is not one-to-one because many horizontal lines cut it twice.

Some functions may be one-to-one only on a restricted domain. For example:

• $f(x)=x^2$ is not one-to-one on $\mathbb{R}$
• but it becomes one-to-one on $[0,\infty)$

This is important in calculus, algebra, and function inversion.

3. Properties and Relation with Inverse Functions

• A function has an inverse function only if it is one-to-one onto its range.
• The inverse exists because each output corresponds to exactly one input.
• If a function is one-to-one, then the inverse relation is also a function.

For example, if $f(x)=3x-1$, then:

$$y=3x-1$$

Solving for $x$:

$$x=\frac{y+1}{3}$$

So the inverse is:

$$f^{-1}(x)=\frac{x+1}{3}$$

This is possible because $f$ is one-to-one.

A useful property:

• If $f$ and $g$ are one-to-one, then their composition $g \circ f$ is also one-to-one.

This makes injective functions very useful in proofs and function construction.

Working / Process

1. Identify the function and its domain and codomain clearly.
2. Assume two arbitrary inputs $a_1$ and $a_2$, and check whether $f(a_1)=f(a_2)$ leads to $a_1=a_2$.
3. If needed, apply the horizontal line test or restrict the domain to make the function one-to-one.

Advantages / Applications

• Helps determine whether an inverse function exists.
• Used in counting and comparing sizes of sets in set theory.
• Useful in solving equations, graph analysis, and theorem proving.
• Important in computer science, cryptography, and coding theory where unique representation matters.
• Used in proving that two sets have the same cardinality through one-to-one correspondence.

Summary

• A one-to-one function maps distinct inputs to distinct outputs.
• It is also called an injective function.
• It can be tested algebraically or by the horizontal line test.
• It is essential for inverse functions and set comparisons.