one one or many one 🤯🤯| relations and function 😍😍 | class 12 ncert 🔥🔥 #class12maths #youtube

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Definition of One-to-One Functions
A function has many types, and one of the most common functions used is the one-to-one function or injective function. Also, we will be learning here the inverse of this function.

One to One Function

One-to-One functions define that each element of one set, say Set (A) is mapped with a unique element of another set, say, Set (B).

Or

It could be defined as each element of Set A has a unique element on Set B.

Or

An injective function (injection) or one-to-one function is a function that maps distinct elements of its domain to distinct elements of its codomain.

In brief, let us consider ‘f’ is a function whose domain is set A. The function is said to be injective if for all x and y in A,

Whenever f(x)=f(y), then x=y

And equivalently, if x ≠ y, then f(x) ≠ f(y)

Formally, it is stated as, if f(x) = f(y) implies x=y, then f is one-to-one mapped, or f is 1-1.

Similarly, if “f” is a function which is one to one, with domain A and range B, then the inverse of function f is given by;

f-1(y) = x ; if and only if f(x) = y

A function f : X → Y is said to be one to one (or injective function), if the images of distinct elements of X under f are distinct, i.e., for every x1 , x2 ∈ X, f(x1 ) = f(x2 ) implies x1 = x2 . Otherwise, it is called many to one function.
Injective function

In Maths, an injective function or injection or one-one function is a function that comprises individuality that never maps discrete elements of its domain to the equivalent element of its codomain. We can say, every element of the codomain is the image of only one element of its domain.

Examples
Examples of Injective Function

The identity function X → X is always injective.
If function f: R→ R, then f(x) = 2x is injective.
If function f: R→ R, then f(x) = 2x+1 is injective.
If function f: R→ R, then f(x) = x2 is not an injective function, because here if x = -1, then f(-1) = 1 = f(1). Hence, the element of codomain is not discrete here.

One to One Graph – Horizontal Line Test
An injective function can be determined by the horizontal line test or geometric test.

If a horizontal line intersects the graph of the function, more than one time, then the function is not mapped as one-to-one.
If a horizontal line can intersect the graph of the function only a single time, then the function is mapped as one-to-one.

Exponential graph

Is Parabola a one to one function?
No, a parabola is not a 1-1 function. It can be proved by the horizontal line test.

A parabola is represented by the function f(x) = x2

Now, if we draw the horizontal lines, then it will intersect the parabola at two points in the graph. Hence, for each value of x, there will be two outputs for a single input.

One to One Function Inverse
If f is a function defined as y = f(x), then the inverse function of f is x = f -1(y) i.e. f-1 defined from y to x. In the inverse function, the co-domain of f is the domain of f -1 and the domain of f is the co-domain of f -1.

Only one-to-one functions have its inverse since these functions have one to one correspondences, i.e. each element from the range corresponds to one and only one domain element.

Let a function f: A - B is defined, then f is said to be invertible if there exists a function g: B - A in such a way that if we operate f{g(x)} or g{f(x)} we get the starting point or value.

Properties of One-One Function
If f and g are both one to one, then f ∘ g follows injectivity.
If g ∘ f is one to one, then function f is one to one, but function g may not be.
f: X → Y is one-one, if and only if, given any functions g, h : P → X whenever f ∘ g = f ∘ h, then g = h. In other words, one-one functions are exactly the monomorphisms in the category set of sets.
If f: X → Y is one-one and P is a subset of X, then f-1 (f(A)) = P. Thus, P can be retrieved from its image f(P).
If f: X → Y is one-one and P and Q are both subsets of X, then f(P ∩ Q) = f(P) ∩ f(Q).
If both X and Y are limited with the same number of elements, then f: X → Y is one-one, if and only if f is surjective or onto function.

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