One to One Function

The phrase ‘one to one relationships’ refers to any two items in which one can solely belong to another item. In mathematics, such relationships are known as one to one functions or injective functions. This is because there are either an equal number of items or one item can be coupled only with one other item. A simple daily life instance of one to one function is a person’s name and their seat number in a train. In this article, we will read about one to one functions. 

Definition of One to One Function

An injective function’s definition states that it is a special function that maps each element of the scope to precisely one element of its domain, ensuring that the outputs never repeat. For example, the function g(x) = x – 4 is a one to one function because it returns a different result for each input. Also, the function g(x) = x2 is not a one to one function because it returns 4 when the inputs are 2 and -2. A many to one function is one that does not have a one-to-one relationship.

Solved Questions on One to One Functions

Question 1

Is the function mentioned below a one to one function?

g = {(-1 , 2),(0 , 4),(2 , -4),(5 , 6),(10 , 0)}

Solution

Examine any two possible values in the domain of function g to see if their corresponding outputs differ. As a result, function g is a one to one function.

Question 2

Is the below-mentioned function given by f a one to one function?

f(x) = -x 3 + 3 x 2 – 2

Solution

A graph and the horizontal line test can assist in answering the preceding question.

Because a horizontal line intersects the graph of ‘f’ at three different points, it implies that there are at least three distinct inputs x1, x2, and x3 with the same output Y, indicating that f is not a one to one function.

Question 3

Find all linear functions of the form given:

f(x) = a x + b

where a and b are real numbers such that ‘a’ not equal to zero is a one to one function.

Solution 

We begin with f(A) = f(B) and demonstrate that this leads to a = b.

a(A) + b = a(B) + b

To get a(A) = a(B), add -b to both sides of the equation (B)

Because a is not equal to zero, divide both sides by a to get A = B.

Since we’ve found that f(A) = f(B) leads to A = B, we can say that all linear functions of the form f(x) = an x + b are one to one functions.

A=B

Question 4

Determine that the function mentioned below

f(x) = a (x – h) 2 + k , for x >= h,

where a, h, and k are real numbers such that a, not equal to zero, is a one to one function.

Solution

You can begin with f(A) = f(B)

Now add -k to both sides of the equation to obtain

a (A – h) 2 + k = a (B – h)2 + k

Now, because it is not equal to 0, divide both sides by a

a (A – h)2 = a (B – h)2

The above equation leads to two other equations

(A – h)2 = (B – h)2

(A – h) = (B – h) or (A – h) = – (B – h)

The first equation leads to

A = B

Let’s find the second equation. The domain of f is all values of x such that x >= h. 

This situation becomes x – h >= 0 which in turn leads to A – h >= 0 and B – h >= 0 which means the second equation (A – h) = – (B – h) does not have a solution.

Question 5

Prove the given function is a one to one function.

 f: R→R defined by f(x)=3x+2 

Solution

Take for instance f(x1)=f(x2) , which means 3×1+2=3×2+2. 

Thus, 3×1=3×2 

so x1=x2 .

We have elaborated if f(x1)=f(x2), then x1=x2. So, f is a one to one function. 

Question 6

Illustrate all the rational functions of the given form, where a and b are real numbers, such that a not equal to zero is a one to one function.

f(x) = 1 / (a x + b)

Solution 

Start by writing an equation

 f(A) = f(B)

1 / (a A + b) = 1 / (a B + b)

Now, multiply both sides of the equation by (a A + b)(a B + b), and then simplify the equations. 

a B + b = a A + b

Add -b to both sides

a B = a A

Divide both sides by a to obtain

B = A

The mentioned functions are one to one functions. 

Conclusion

The one to one function maps distinct elements to distinct elements. In other words, an element of a codomain is an image of, at most, one element of the domain. In this article, we talked about one to one function or injective function. We also presented questions for you to practice on.