One to One Function - Definition and Examples

One to One Function

A one to one function, also known as an injective function, is a mathematical rule in which every input has a unique output. In simple words, no two different inputs give the same result. The one to one function definition helps in understanding relationships where each element in the domain (input set) connects to a distinct element in the range (output set).

In mathematics, functions describe how one quantity depends on another. A one to one function is a function where the value of x has a single value of y. The concept plays a significant role in algebra, geometry, calculus, and real life uses like coding, data arrangement and mapping systems. Understanding about what are one to one functions allows students to learn about how functions operate, how to recognize them, and how to find their inverses.

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What are One to One Function?

One to one functions are special types of functions where every input corresponds to a unique output, and every output is linked to one input only. This means there are no repeated results for different inputs.

Mathematically, a function f(x) is one to one if f(a) = f(b) implies a = b. This ensures that each element of the domain maps to exactly one element in the range.

Examples:

• f(x) = x + 5 → one to one function because each input gives a different output.

• f(x) = x² → not one to one because f(2) = f(-2).

The one to one function definition helps identify functions that can have inverses, since only one to one functions have inverses that are also valid functions.

These functions are useful in many fields such as cryptography, data analysis, and computer programming, where unique mapping between elements is required.

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One to One Function Graph

The one to one function graph is a visual way to understand whether a function is injective or not.

To check this, we use the horizontal line test. According to this test, if every horizontal line drawn across the graph cuts it at most once, then the function is one to one.

For example:

• The graph of f(x) = x + 3 passes the horizontal line test, meaning it is a one to one function.

• The graph of f(x) = x² fails this test because some horizontal lines intersect the graph twice, meaning it is not one to one.

The one to one function graph helps students visualize how inputs and outputs relate. It’s a quick and effective way to check if a function has an inverse or if it passes the test of uniqueness.

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How to Determine if a Function is One to One?

There are two main methods to check whether a function is one to one: the graphical method and the algebraic method.

1. Graphical Method (Horizontal Line Test)
Draw the graph of the function and pass horizontal lines across it.

• If a line touches the graph only once → the function is one to one.

• If a line touches more than once → the function is not one to one.

2. Algebraic Method

Assume f(a) = f(b) and prove that a = b.

If this is true for all values, the function is one to one.

Example: 

For f(x) = 3x + 1
f(a) = 3a + 1 and f(b) = 3b + 1.
If f(a) = f(b), then 3a + 1 = 3b + 1 → a = b.

Hence, it is a one to one function.

These methods help determine the one to one function definition clearly and are useful in solving questions related to inverses and mapping in mathematics.

Properties of One to One Function

The properties of one to one function explain the key features that make it unique and easy to identify. These properties help in understanding how each input and output are connected in a special way.

1. Injectivity:

Every input has a special output. There is no single result of some two different inputs. When f(a) = f(b) can result in a = b, it means that the function is one to one.

2. Existence of Inverse:

One to one functions have inverses which are also functions. The inverse is the reverse of the mapping i.e. it transforms the outputs into their original inputs.

3. Horizontal Line Test:

A one to one function passes the horizontal line test, which implies that none of horizontal lines crosses the graph more than once. This indicates that there is only one input value per output value.

4. Domain and Range Relationship:

In a one to one function, each element of the domain (input) matches exactly one element of the range (output). This makes it possible to find an inverse function.

5. Composition Property:

When one has two one to one functions, then the outcome is also one to one function. It is helpful in more complex subjects such as transformations and composition of functions.

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Steps to Find Inverse of One to One Function

One to one functions have inverses which are valid since each output in the range is associated with a single input in the domain. The inverse of a one to one function switches the position of the inputs and outputs, and we are capable of tracing the results of this to the original value.

To get the inverse of a one to one function use the following steps:

Step 1: Replace f(x) with y

Write the function as y = f(x) to make it easier to work with.

Step 2: Swap the variables

Exchange x and y to get x = f(y). This shows that you are reversing the function.

Step 3: Solve for y

Rearrange the equation so that y is by itself on one side. This expresses the inverse relationship.

Step 4: Write the inverse function

Replace y with f⁻¹(x) to represent the inverse function.

Example: Given f(x) = 2x + 3

Step 1: Write as y = 2x + 3

Step 2: Swap → x = 2y + 3

Step 3: Solve → y = (x - 3)/2

Step 4: Write inverse → f⁻¹(x) = (x - 3)/2

This shows that the inverse function undoes the operation of the original function. If f(x) doubles and adds 3, then f⁻¹(x) subtracts 3 and divides by 2.

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