One to One Functions

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. A one to one function, also known as an injective function, is a specific type of function with a unique characteristic: each element of the domain (input) is mapped to a distinct element of the codomain (output). This means that no two different elements in the domain have the same image in the codomain.

Definition

A function $f: A \rightarrow B$ is called one to one (injective) if for every $x_1, x_2 \in A$, whenever $f(x_1) = f(x_2)$, it implies that $x_1 = x_2$.

In other words, different inputs map to different outputs.

Mathematical Representation

If $f$ is a function from set $A$ to set $B$, then $f$ is one to one if:

$$ \forall x_1, x_2 \in A, f(x_1) = f(x_2) \Rightarrow x_1 = x_2 $$

Alternatively, the contrapositive statement is also used to define a one to one function:

$$ \forall x_1, x_2 \in A, x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2) $$

Horizontal Line Test

A graphical method to determine if a function is one to one is the horizontal line test. If any horizontal line intersects the graph of the function at most once, then the function is one to one.

Differences and Important Points

Property	One to One Function	Not One to One Function
Definition	Each element of the domain maps to a unique element in the codomain.	At least one element of the domain maps to the same element in the codomain as another element.
Mathematical Test	$f(x_1) = f(x_2) \Rightarrow x_1 = x_2$	There exists $x_1 \neq x_2$ such that $f(x_1) = f(x_2)$
Horizontal Line Test	Any horizontal line intersects the graph at most once.	Some horizontal lines intersect the graph more than once.
Injectivity	Yes	No

Examples

Example 1: One to One Function

Consider the function $f(x) = 2x + 3$. Let's check if it is one to one.

Take any $x_1, x_2 \in \mathbb{R}$, and suppose $f(x_1) = f(x_2)$. This gives us:

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

Subtracting 3 from both sides and then dividing by 2, we get:

$$ x_1 = x_2 $$

Since $x_1$ must equal $x_2$ whenever $f(x_1) = f(x_2)$, the function $f(x) = 2x + 3$ is one to one.

Example 2: Not One to One Function

Now, consider the function $g(x) = x^2$. Let's check its injectivity.

Take $x_1 = 1$ and $x_2 = -1$, we find that:

$$ g(1) = 1^2 = 1 $$ $$ g(-1) = (-1)^2 = 1 $$

Here, $g(x_1) = g(x_2)$ but $x_1 \neq x_2$. Therefore, the function $g(x) = x^2$ is not one to one.

Example 3: Using the Horizontal Line Test

Let's use the horizontal line test on the function $h(x) = \ln(x)$.

The graph of $h(x) = \ln(x)$ is such that any horizontal line will intersect it at exactly one point. Therefore, by the horizontal line test, $h(x)$ is a one to one function.

Conclusion

One to one functions are fundamental in mathematics, especially when discussing function inverses, as only one to one functions have inverses that are also functions. Understanding the concept of injectivity and being able to determine whether a function is one to one is crucial for various fields of mathematics, including calculus, algebra, and more.