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Introduction To A 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.
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FAQs on One to One Function
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).