One to One Function

Before getting to know what exactly A One-to-One Function (also written as 1 to 1) is, let's talk about a function. So, a function is to say that it provides an output for an allotted input. You provide functions with a definite value to start with and they carry out their operation on the value, and then they submit you the answer. For instance, the function f(x) = x + 2 adds 2 to any value you allotted. You feed it a 7, this function will give you a 9: f (7) = 7 + 2 = 6. Likewise, a one-to-one function is that form of a function in which the answers never repeat.

One-to-One Function Explained

While an ordinary function can possess two different input values that yield the same answer, but a one-to-one function will never. For instance, the function f(x) = x^2 is not a one-to-one function that’s simply because it yields an answer 4 when you input both a 2 and a -2, also you can refer as many to one function. But the function f(x) = x - 3 is 1 to 1 since it brings forth a distinctive answer for every input.

One-to-One Function and Its Inverses

The statement "A Chinese restaurant sells Hakka noodles" could be perceived as a function. You can even plot it on a graph, with various restaurants along the x-axis, and various foods the restaurant specializes in on the y-axis. Any time a Chinese hakka noodle restaurant is input into the function, it would output "hakka noodle" as the specialized food.

Whether or not this Chinese restaurant has a one-to-one function? How can we show this?

“Chinese restaurants sell Hakka noodles" is a function. But, it is NOT a one-to-one function.

For the purpose of it to be one-to-one, it should be invertible, giving something like: "Hakka noodle sellers are Chinese restaurants", and that statement should also be a function since bakeries, super markets also sell Hakka noodles.

Besides, a quick and correct test for determining a one-to-one function is the horizontal line test. If a horizontal line is seen to bisect the graph of the function at more than one place, the function is clearly NOT one-to-one.

Horizontal Line Test

A function {f} is one-to-one and also has an inverse function if and only if no horizontal line bisects the graph of f in more than one point.

To prove the horizontal test. Lets first plot a graph of the function , one like below 

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Now, secondly Let's use  the Horizontal Line Test.

Imagine a multitude of  horizontal lines and see through places where the graph is bisected more than once.

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Since, No horizontal line is seen to bisect the graph in more than one place and hence this shows the function to have an inverse.

Now scout through the concept of One-to-One Function using examples. 

Solved Examples 

Example 1

Problem:

Prove arithmetically that all linear functions of the algebraic expression f(x) = a x + b , with a ≠ 0, are 1 to 1 functions.

Solution

First, we will apply the contra positive that clearly describes that function f is a 1 to 1 in case the following is true:

On condition that, f(x1) = f(x2) then x1 = x2

Now, we begin with

f {x1} = f {x2}

Which assigns us – {A x1 + b} = {a x2 + b}

Simplify to get

A (x1 - x2) = 0

Seeing that, a ≠ 0,

Now the only condition for the aforementioned to be satisfied is to have

X1 - x2 = 0

That assigns

X1 = x2

We have presented that f(x1) = f(x2) that results in x1 = x2 and as per the contra positive above, all linear functions of the expression f(x) = a x + b, with a ≠ 0, are 1 to 1 functions.

Example 2

Problem:

Prove graphically and rationally that the linear function f(x) = - x2 + 3 is “NOT” a 1 to 1 function

Solution

Yet again, we will use the contra positive that clearly describes that function f is a 1 to 1 in case the hereunder is true:

On a condition that, f {x1} = f{x2} then, x1 = x2

to begin with,

F {x1} = f{x2}

this assigns us –  - {x12 + 3} = - {x22 + 3}

Now, Simplify to get,

- {x12 - x22} = 0

Factor - (x1 - x2)( x1 + x2) = 0 which assigns  x1 - x2 = 0 or x1 + x2 = 0

Which further provides,  x1 = x2 or x1 = - x2

Seeing that the above experiment does not critically draw the inference that x1 = x2 the function is not a 1 to 1 function.

Underneath is a graph of the allotted function with 2 two horizontal lines drawn: (x axis and y-axis) = - 2 (slanting line) that clearly depicts that there are two points of bisections and hence does not make for a 1 to 1 function.

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Fun Facts

It is not true that all functions have inverse functions.