One to One Function

Unlike regular functions, which can have multiple input values producing the same output, one-to-one functions do not allow this. For different inputs, the outputs corresponding to them are also different. Let's begin by exploring the definition and properties of one-to-one functions. 

The function in which each output value is paired with a unique input is a one-to-one function. These functions ensure that two different inputs never produce the same output. 

For example,

let \(f (x) = x + 3\)

Solution: 
We evaluate the function at different inputs
  \(f (1) = 1 + 3 = 4\)
  \(f (-1) = -1 + 3 = 2\)
  \(f (2) = 2 + 3 = 5\)
  \(f (4) = 4 + 3 = 7\)

This shows that each input gives a different output. However, to prove that \({f(x) = x + 3}\) is a one-to-one function, we need to show that if, \({f({x_{1}}) = f({x_{2}})}\), then, \({{x_{1}} = {x_{2}}}\) 

Let’s assume that, \({{f{(x_{1})} = {f({x_{2})}} \implies {x_{1}} + 3 = {x_{2}} + 3 \implies {x_{1}} = {x_{2}}}}\)

Hence, it is proved that two different inputs do not map to the same output; so, \({f(x) = x + 3}\) is injective or a one-to-one function.
 

A one-to-one function, also known as an injective function, is characterized by its ability to map well-defined elements of its domain to definite elements of its co-domain, and here are some key properties that help us understand its characteristics:

1. A function \(f: A → B\) is one-to-one if distinct elements in the domain A are mapped to distinct elements in the co-domain B. Formally: If \({f{(x_{1})}} = {f({x_{2}})}\), then \({x_{1}} = {x_{2}}\).

2. A function is one-to-one only if every horizontal line intersects its graph once at most. This is known as the horizontal line test.

3. Given a one-to-one function, we can define its inverse, \({f^{-1}}\), which is characterized by the following property:
\({{f^{-1}}{(f(x))}} = {x}\), for all x in the domain of f
\({f} {({f^{-1}}(y))} = {y}\), for all y in the domain of \({f^{-1}}\).

4. In the context of real functions, a significant number of one-to-one functions are monotonic in a strict sense. This implies they are either consistently increasing, meaning larger inputs always produce larger outputs \({{({x_{1}} < {x_{2}} → {f{(x_{1})}} < {f({x_{2}}))}}}\), or consistently decreasing, where a larger input always results in a smaller output
 \({({x_{1}} < {x_{2}} → {f{(x_{1})}} > {f{(x_{2})}}}\).

5. It's important to note that injectivity and surjectivity are separate concepts in the study of functions. A function can either be injective or surjective but not both.

6. For a one-to-one function f from a finite set A to a finite set B, the number of elements in A must be less than or equal to the number of elements in B. This can be written as |A|  |B|.

Below is the image of a one-to-one function and inverse function 
 

A function qualifies as one-to-one when all horizontal lines cross its graph only once. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.

A horizontal line is defined by a y-value that remains the same. The line intersects the graph at two or more points means \({{f{(x_{1})}} = {f{(x_{2})}}}\) for \({x_{1} \ne  x_{2}}\). 

The definition of one-to-one is violated, which requires
 \({{f{(x_{1})}}= {f{(x_{2})}} \implies  {x_{1}} = {x_{2}}}\)

For example: 
\(f(x) = {x^{2}}\) is not one-to-one because a horizontal line like \(y = 4\) intersects the graph at \(x = {-2}\) and \(x = 2\), but,
\({f(x) = x^{3}}\) is one-to-one because every horizontal line intersects the curve of \(f(x) = {x^{3}}\) at most once, satisfying the condition for horizontal line test.
 

The horizontal line test is used to determine whether a given relation is a function, and there are two methods to determine if a function is one-to-one.
 

• Tested Graphically, A function is considered one-to-one if its graph passes the horizontal line test, meaning it intersects the graph at a unique value of y for every x.

• To test if a function g is one-to-one algebraically, assume \(g(a) = g(b)\) and check if it leads to a = b. If it does, the function is one-to-one.

• A function g(x) is one-to-one if its derivative is either exclusively positive or negative throughout its domain, meaning it is either entirely increasing or decreasing. For example, the exponential function \({f(x)} = {e^x}\) is strictly increasing across its entire domain \((-∞,∞ )\), which makes it one-to-one. This can be verified by observing the graph.

All functions can be represented in a graphical form. A one-to-one function is represented on a Cartesian plane using a line or a curve on a plane as per the Cartesian system. The domain is marked horizontally with respect to the x-axis, and the range is marked vertically in the direction to the y-axis. For a one-to-one function g, no two points \({({x_{1}}, {y_{1}})}\) and \({({x_{2}}, {y_{2}})}\) will have the same y-value. A one-to-one function is a function in which every y-value is paired with exactly one x-value, meaning it never takes on the same y-value more than once. 

To understand inverse functions, it is important to be aware of one-to-one functions. A one-to-one function's inverse function gives back the input and output values of the original function. In other words, if a one-to-one function assigns x to y, its inverse function assigns y to x. The inverse of a one-to-one function g is written as \({g^{-1}}\)^. If it is one-to-one, you can find the pairs for \({g^{-1}}\) by simply flipping the input and output of each pair in g. What was the input for g becomes the output for \({g^{-1}}\), and what was the output for g becomes the input for \({g^{-1}}\).

Now that we have discussed what is the inverse of a one-to-one function is, in this section, we will learn about the key properties that define it.

1. The inverse function is the reverse function of the original function. If f is one-to-one and has an
inverse \({f^{-1}}\), then 
Applying \(f\) and then \({f^{-1}}\) to any valid input for \({f^{-1}}\) gets you back to that original input.

Applying \({f^{-1}}\) and then \(f\) to any valid input for \(f\) also gets you back to that original input.

2. If a one-to-one function \(f\) takes values from set A and produces values in set B, this is written as
\(f: A → B\), then its inverse function, \({f^{-1}}\), does the opposite. It takes values from set B and produces values in set \(​ A ( {f^{-1}}: B → A) ​\). The input and output sets are simply switched.

3. If a function f is one-to-one (injective), then its inverse \({f^{-1}}\) is also one-to-one.

4. The graph of \({f^{-1}}\) is the reflection of the graph of f across the line \(y = x\).

5. Two functions, f and g, are inverses if and only if 
The composition of f with g, written as \((f \cdot g) (x)\), equals x for all x in the domain of g, 
The composition of g with f, written as \((g \cdot f) (x)\), equals x for all x  in the domain of f.

This suggests that each function reverses the effect of the other.

Here is a graph that aligns with the properties of inverse functions

To find the original values of a one-to-one function, we apply the inverse of a one-to-one function. To find the inverse \({f^{-1}}\) of a one-to-one function, follow these steps;

Rewrite: Start by simply writing down the function.

Interchanging: Interchange x and y in the equation, resulting in \(x = f(y)\). This shows that inputs and outputs reverse in the inverse function.

The new equation for y: Rearrange the new equation to solve for y. This often involves using inverse mathematical operations.

Replace y with \({{f^{-1}}(x)}\): Once you have Y by itself, replace it with the notation \({{f^{-1}}{(x)}}\) to represent the inverse function.

 

For Example, let's find the inverse of \(f(x) = 3x + 5\).
 

Step 1: Starting with \(y = 3x + 5\).
 

Step 2: Interchange x and y, so \(x = 3y + 5\).
 

Step 3: Solve for Y:
 

• Subtract 5 from both sides: \(x - 5 = 3y\).
   
• Divide by 5: \({{y} = {{x - 5} \over 3}}\)
   

Step 4: The inverse function is \({f^{-1}{(x)}} = {{x-5}\over {3}}\).
 

A one-to-one function is an important concept in mathematics, especially in algebra, calculus, and precalculus. Understanding these tips and tricks helps students to master them.

• Remember that for a one-to-one function, each element in the domain is mapped to a unique element in the co-domain.
   
• To check if a function is one-to-one, use a horizontal line. Draw a horizontal line on the graph. If the line touches the graph more than once, then the function is not one-to-one.
   
• Use the inverse test, as a one-to-one function always has an inverse. To confirm, check both directions: \({{f{(f^{-1}{x})}} = x}\) and \({{f^{-1}{({f(x)})}} = x}\). If both the functions are correct, then the inverse function is correct. 
   
• Understand the notation, \({f^{-1}{(x)}} \) is the inverse function and \({{1} \over {f{(x)}}}\) is the reciprocal of the function. 
   
• Remember that if \({f{(x_{1})} = {f{(x_{2})}}}\), and when solving it gives \({x_{1}} = {x_{2}}\), then the function is one-to-one.

One-to-one functions are used in many fields of daily life because each input has a unique output. Here are a few real-life applications.

• In cryptography, one-to-one functions are used to encode and decode messages. Each original message is mapped to a unique coded message to ensure correct decoding. For example, if \({{E{(x)} = {x + 5}}}\) and the message is 10, it becomes 15. To decode, the inverse function \({{{D{x}} = {x - 5}}}\) gives back 10. This ensures that every code can be reinterpreted to retrieve the exact original message. 
   
• In finance and banking, the compound interest formula \({{A = {P{(1 + r)}^{t}}}}\) is a one-to-one function in terms of time t. Using the inverse function, we can calculate how long it will take to reach any desired amount of money. 
   
• In data compression, a one-to-one function is used to store and transmit information effectively. In data compression algorithms like Huffman coding, each unique piece of data maps to a distinct code, so the original data can be recovered.
   
• A one-to-one function is used in ID systems, so each person or item has a unique identifier. For example, student ID 2025A123 belongs to only one student, making identification easy.
   
• In medical imagining, one-to-one functions are used to convert signal data into distinct images. Each unique data set creates a unique image, allowing doctors to interpret results accurately. The process is reversible in theory, ensuring precise diagnosis.