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One to one function basically denotes the mapping of two sets. A function g is one-to-one if every element of the range of g corresponds to exactly one element of the domain of g. One-to-one is also written as 1-1. A function f is 1 -to- 1 if no two elements in the domain of f correspond to the same element in the range of f . In other words, each x in the domain has exactly one image in the range.

This means that given any x, there is only one y that can be paired with that x. A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b. All elements in B are used. A graph of a function can also be used to determine whether a function is one-to-one using the horizontal line test: If each horizontal line crosses the graph of a function at no more than one point, then the function is one-to-one.

The horizontal line test can be used to determine if a function is a one-to-one given a graph. Simply superimpose a horizontal line onto a graph and see if it intersects the graph at more than one point. If it does, the graph is not one-to-one and if it only intersects at one point, it will be one-to-one. If some horizontal line intersects the graph of the function more than once, then the function is not one-to-one. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one.

Testing one to one function graphically: If the graph of g(x) passes through a unique value of y every time, then the function is said to be one to one function (horizontal line test). Testing one to one function algebraically: The function g is said to be one to one if a = b for every g(a) = g(b). It is essential for one to understand the concept of one-to-one functions in order to understand the concept of inverse functions and to solve certain types of equations. Firstly, a function g has an inverse function, g-1, if and only if g is one to one. In the below-given image, the inverse of a one-to-one function g is denoted by g−1, where the ordered pairs of g-1 are obtained by interchanging the coordinates in each ordered pair of g. Here the domain of g becomes the range of -1, and the range of g becomes the domain of g-1.

In order for function to be a one to one function, g( x_{1} ) = g( x_{2} ) if and only if x_{1} = x_{2} . Let us start solving now:

g( x_{1} ) = -3 x_{1}^{3} – 1

g( x_{2} ) = -3 x_{2}^{3} – 1

We will start with g( x_{1} ) = g( x_{2} ). Then

-3 x_{1}^{3} – 1 = -3 x_{2}^{3} – 1

-3 x_{1}^{3} = -3 x_{2}^{3}

( x_{1} )^{3} = ( x_{2} )^{3}

Removing the cube roots from both sides of the equation will lead us to x_{1} = x_{2}.

Answer:** **Hence, g(x) = -3x^{3} – 1 is a one to one function.

A graph of a function can also be used to determine whether a function is one-to-one using the horizontal line test: If each horizontal line crosses the graph of a function at no more than one point, then the function is one-to-one. One to one function basically denotes the mapping of two sets. A function g is one-to-one if every element of the range of g corresponds to exactly one element of the domain of g. One-to-one is also written as 1-1. A function f() is a method, which relates the elements/values of one variable to the elements/values of another variable, in such a way that the elements of the first variable identically determine the elements of the second variable.

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 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 having 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 that 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

Theorem If f is a one-to-one continuous function defined on an interval, then its inverse f−1 is also one-to-one and continuous. (Thus f−1(x) has an inverse, which has to be f(x), by the equivalence of equations given in the definition of the inverse function.). If a horizontal line intersects the graph of the function in more than one place, the function is not one-to-one

Every one-to-one function f has an inverse; this inverse is denoted by f−1 and read aloud as 'f inverse'. A function and its inverse 'undo' each other: one function does something, the other undoes it. Two functions f and g are inverse functions if and only if: (1) For all x in the domain of f, g(f(x)) = x, and (2) for all x in the domain of g, f(g(x)) = x.

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