This paper introduces the definitions of the mode function and the graph of a function, defining the graph of a function as identical with the function. It defines several concepts including the domain, range, identity function, composition of functions, one-to-one function, inverse function, restriction of a function, image, and inverse image. Basic facts about functions and these concepts are also proved. The notation and terminology are introduced in papers [1] and [2]. The paper defines the mode function, which is a function that maps any element to any other element. It then defines the graph of a function as the set of ordered pairs of elements from the function. Several propositions are stated, including that the graph of a function is equal to the function itself, and that the graph of a function is uniquely determined by its elements. The paper also defines the domain and range of a function, and proves several propositions about them. It defines the composition of functions, and proves several properties of function composition. It defines the identity function, and proves several properties of the identity function. The paper also defines the inverse function, and proves several properties of the inverse function. It defines the restriction of a function, and proves several properties of the restriction of a function. It defines the image and inverse image of a function, and proves several properties of the image and inverse image. The paper also defines several schemes, including the scheme GraphFunc and the scheme FuncEx, which are used to define functions based on certain conditions. It also defines the scheme Lambda, which is used to define functions based on a unary functor. The paper then defines several propositions about functions, including that a function is one-to-one if and only if it is injective, and that the composition of two one-to-one functions is also one-to-one. It also defines the inverse of a function, and proves several properties of the inverse of a function. The paper also defines the restriction of a function to a set, and proves several properties of the restriction of a function. It defines the image and inverse image of a function, and proves several properties of the image and inverse image. The paper concludes by stating several propositions about functions, including that the inverse of a function is unique, and that the composition of functions is associative. It also proves several properties of the inverse of a function, including that the inverse of the inverse of a function is the function itself.This paper introduces the definitions of the mode function and the graph of a function, defining the graph of a function as identical with the function. It defines several concepts including the domain, range, identity function, composition of functions, one-to-one function, inverse function, restriction of a function, image, and inverse image. Basic facts about functions and these concepts are also proved. The notation and terminology are introduced in papers [1] and [2]. The paper defines the mode function, which is a function that maps any element to any other element. It then defines the graph of a function as the set of ordered pairs of elements from the function. Several propositions are stated, including that the graph of a function is equal to the function itself, and that the graph of a function is uniquely determined by its elements. The paper also defines the domain and range of a function, and proves several propositions about them. It defines the composition of functions, and proves several properties of function composition. It defines the identity function, and proves several properties of the identity function. The paper also defines the inverse function, and proves several properties of the inverse function. It defines the restriction of a function, and proves several properties of the restriction of a function. It defines the image and inverse image of a function, and proves several properties of the image and inverse image. The paper also defines several schemes, including the scheme GraphFunc and the scheme FuncEx, which are used to define functions based on certain conditions. It also defines the scheme Lambda, which is used to define functions based on a unary functor. The paper then defines several propositions about functions, including that a function is one-to-one if and only if it is injective, and that the composition of two one-to-one functions is also one-to-one. It also defines the inverse of a function, and proves several properties of the inverse of a function. The paper also defines the restriction of a function to a set, and proves several properties of the restriction of a function. It defines the image and inverse image of a function, and proves several properties of the image and inverse image. The paper concludes by stating several propositions about functions, including that the inverse of a function is unique, and that the composition of functions is associative. It also proves several properties of the inverse of a function, including that the inverse of the inverse of a function is the function itself.