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.

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.

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.

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.

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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It is not true that all functions have inverse functions.

FAQ (Frequently Asked Questions)

1. What is the application of One to One Functions?

One to one functions can be used in varied ways such as:-

ü Solving definite kinds of mathematical equations

ü Solving certain types of equations using logarithms

ü Using inverse functions that also have One-to-One Function (though not always)

2. How can we express a function?

There are different ways in which a function can be expressed such as:-

ü A function can be conveyed in the form of a list, particularly if the domain and range are small.

ü It can be conveyed in the form of a graph. The function is characterized by a curve plotted on a cartesian plane, in which the domain is drawn horizontally (along x-axis) while the range is drawn vertically (along y-axis).

ü A function can also be conveyed by way of formula. The formula is used to compute the range value for any assigned domain value.

3. What is the Inverse of a function?

We know that the inverse of addition is subtraction and the inverse of multiplication is division. Then what about the inverse of a function? An inverse of a function is that which undoes the action of another function. That is to say, function ‘h’ is the inverse of a function ‘f’ whenever y= f (x) then x= h (y).