Inverse Function Theorem

The inverse function theorem in mathematics, specifically differential calculus, states that a function is invertible in the neighborhood of a point in its domain if its derivative is continuous and non-zero at the point. The theorem also provides a formula for the inverse function’s derivative. This theorem can be generalized in multivariable calculus to any continuously differentiable, vector-valued function whose Jacobian determinant is non-zero at a point in its domain, yielding a formula for the Jacobian matrix of the inverse. The inverse function theorem can also be applied to complex holomorphic functions, differentiable maps between manifolds, differentiable functions between Banach spaces, and so on.

The inverse of a function and its derivation are determined by the type of problem presented to us. Before proceeding to the inverse function theorem, it is critical to first understand what is inverse of a function.

Definition of inverse function

“In mathematics, an inverse is a function that serves to “undo” another function. That is, if f(x) produces y, then putting y into the inverse of f yields x. An invertible function f is one that has an inverse, and the inverse is denoted by f^-1“.

The domain of the inverse function is formed by the range of the inverse function and the range of the given function.

By swapping the domain and range of the given function, the domain and range of an inverse function can be obtained. The range of the given function becomes the domain of the given function, and the domain of the given function becomes the range of the inverse function.

The domain value of x is obtained by combining the function f and the reciprocal function f^-1.

Inverse function theorem

If f is a smooth function, then f:Rⁿ→Rⁿ

If the Jacobian is invertible at 0, then there is a neighborhood U with 0 in it.

f:U→f(U)  Denotes a diffeomorphism. In other words, there is a smooth inverse.

f^-1:f(U)→U

Only one-to-one functions are covered by the inverse function theorem. In solving complex inverse trigonometric and graphical functions, the inverse function theorem is used. 

If and only if two functions, f and g, are inverse functions, then:

For all x in f’s domain, g(f(x)) = x.

For all x in g’s domain, f(g(x)) = x.

PROOF

The “only if” part is as follows: Assume f and g are inverse functions. In the function f, let (a, b) be any ordered pair. Then f(a) equals b. The ordered pairs in g are the inverse of those in f, so (b, a) is an ordered pair in the function g, according to the definition of inverse. As a result, g(b) = a. Now consider the composites.

1. g(f(a))=g(b) =a for any number an in the domain of f.

2. f(g(b))=f(a) =b for any number b in the domain of g.

The “if” clause: Assume (1) and (2) in the theorem’s statement are true. Let (a, b) represent any point on the function f. As a result, f(a) = b, and g(f(a))=g(b). However, using (1), g(f(a)) = a, so g(b) = a via transitivity. This indicates that (b, a) belongs to the function g. As a result, g includes all of the points obtained by reversing the coordinates in f.

We can use the same logic to show that f contains all points obtained by reversing the coordinates of g. As a result, f and g are inverse functions.

Characteristics inverse of function theorem

Inverse function characteristics include 

Every one-to-one function f has an inverse, denoted by f^-1 and read aloud as ‘f ‘inverse.’ A function and its inverse undo each other: one function does something, and the other reverses it.

A function and its inverse ‘undo’ each other: one function does something, and the other reverses it.

Conclusion

We conclude in this article that every one-to-one function f has an inverse, which is denoted by f^-1 and read aloud as ‘f’ inverse.’ A function and its inverse ‘undo’ each other: one function does something, and the other reverses it. The inverse function theorem states that if a function is a continuously differentiable function, i.e., the variable of the function can be differentiated at each point in the domain of, then the inverse function is also a continuously differentiable function, and the derivative of the inverse function is the reciprocal of the derivative of the original function. In differential calculus, the inverse function theorem states that a function is invertible in the neighborhood of a point in its domain if its derivative is continuous and non-zero at the point.