Composition of Functions: The composition of functions, also known as function composition, is a new function that is formed by applying a function to the o...
Composition of Functions:
The composition of functions, also known as function composition, is a new function that is formed by applying a function to the output of another function. The new function takes a function as input and outputs a function. In other words, the output of the new function is the input of the original function.
Example:
Let f(x) = x + 2 and g(x) = x - 3. Then (f∘g)(x) = (x + 2)(x - 3) = x^2 - x - 6.
Invertibility:
A function is invertible if it has a unique inverse function. An inverse function is a function that undoes the effect of the original function. In other words, if f(x) = y, then y = f^(-1)(x).
Important Property of Invertibility:
An invertible function is bijective, meaning that each element in the domain maps to a unique element in the range, and each element in the range is mapped to by exactly one element in the domain.
Invertibility and Composition:
If a function is invertible, then its composition with another invertible function is also invertible. This means that (f∘g)'(x) = (f∘g)'(x) = (f'(x))(g'(x)).
Examples:
The function f(x) = x^2 is invertible, with the inverse function f^(-1)(x) = √x.
The function g(x) = x + 1 is invertible, with the inverse function g^(-1)(x) = x - 1.
The function h(x) = x + 2 is invertible, with the inverse function h^(-1)(x) = x - 2