 Hi and welcome to the session, today we will learn about types of functions. So first of all let us see what is a function let A and B be two non-empty sets then a relation from A to B is called a function if every element has a unique image in B and this is denoted by f is a function from A to B. Now let us move on to types of functions first one we have one one or injective function a function f from x to y is defined to be one one or injective if the images of distinct elements under the function f are distinct that is x one x two belonging to x f of x one equal to f of x two implies that x one is equal to x two otherwise one that is for many elements they can be one image now next we have onto or surjective function a function from x to y is said to be onto or surjective every element is the image some element that is for every y belonging to y there exists an element f of x is equal to y and now third we have bijective function a function x to y is said to be bijective if it is both one one let us take one example now here we are given the function f from r to r such that f of x is equal to three minus four x and we need to show that f is bijective now to show that f is bijective we need to show that f is one one and onto so first of all it is show that f is one one for this we will show that for every x one x two f of x one equal to f of x two implies that x one is equal to x two so here let us take f of x one equal to f of x two so this implies now f of x is three minus four x so f of x one will be three minus four x one equal to three minus four x two so this implies minus four x one is equal to minus four x two which gives us x one is equal to x two so from here we get therefore f is one one now we need to show that f is onto and for that we will show that for every y belonging to r there exists some x belonging to r such that f of x is equal to y so let y is equal to three minus four x so from here we get x is equal to three minus y upon four thus for each y belonging to r there exists x equal to three minus y upon four belonging to r such that f of x is equal to f of three minus y upon four which is equal to three minus four into three minus y upon four which is equal to y so for y belonging to r we have x belonging to r such that f of x is equal to y therefore we get f is onto now we have f is one one and f is onto so from here we can say that f is bijective so in this session we have learned types of functions and with this we finished this session hope you must have understood all the concepts goodbye take care and have a nice day