The notion of a function along with some special functions like identity function, constant function, polynomial function, rational function, modulus function, signum function etc. along with their graphs are known to you as you have studied them in previous chapter.

Addition, subtraction, multiplication and division of two functions have also been studied. As the concept of function is of paramount importance in mathematics and among other disciplines as well, we would like to extend our study about function from where we finished earlier. In this topic, we would like to study different types of functions.

Consider the functions f_{1}, f_{2}, f_{3} and f_{4} given by the following diagrams.

In Fig 1.2, we observe that the images of distinct elements of X_{1} under the function f_{1} are distinct, but the image of two distinct elements 1 and 2 of X_{1} under f_{2} is same, namely b. Further, there are some elements like e and f in X_{2} which are not images of any element of X_{1} under f_{1}, while all elements of X_{3} are images of some elements of X_{1} under f_{3}. The above observations lead to the following definitions:

Definition A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x_{1}, x_{2} ∈ X, f (x_{1}) = f (x_{2}) implies x_{1} = x_{2}. Otherwise, f is called many-one.

The function f_{1} and f_{4} in Fig 1.2 (i) and (iv) are one-one and the function f_{2} and f_{3} in Fig 1.2 (ii) and (iii) are many-one.

Definition A function f : X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f (x) = y.

The function f_{3} and f_{4} in Fig 1.2 (iii), (iv) are onto and the function f_{1} in Fig 1.2 (i) is not onto as elements e, f in X_{2} are not the image of any element in X_{1} under f_{1}.

Remark f : X → Y is onto if and only if Range of f = Y.

Definition A function f : X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto.

The function f_{4} in Fig 1.2 (iv) is one-one and onto.

Example Show that the function f : N → N, given by f (x) = 2x, is one-one but not onto.

Solution The function f is one-one, for f (x_{1}) = f (x_{2}) ⇒ 2x_{1} = 2x_{2} ⇒ x_{1} = x_{2}. Further, f is not onto, as for 1 ∈ N, there does not exist any x in N such that f (x) = 2x = 1.

Example Prove that the function f : R → R, given by f (x) = 2x, is one-one and onto.

Solution f is one-one, as f (x_{1}) = f (x_{2}) ⇒ 2x_{1} = 2x_{2} ⇒ x_{1} = x_{2}. Also, given any real number y in R, there exists y/2 in R such that f (y/2) = 2 . (y/2) = y. Hence, f is onto.

Example Show that the function f : N → N, given by f (1) = f (2) = 1 and f (x) = x – 1, for every x > 2, is onto but not one-one.

Solution f is not one-one, as f (1) = f (2) = 1. But f is onto, as given any y ∈ N, y ≠ 1, we can choose x as y + 1 such that f (y + 1) = y + 1 – 1 = y. Also for 1 ∈ N, we have f (1) = 1.

Example Show that the function f : R → R, defined as f (x) = x^{2}, is neither one-one nor onto.

Solution Since f (– 1) = 1 = f (1), f is not oneone. Also, the element – 2 in the co-domain R is not image of any element x in the domain R (Why?). Therefore f is not onto.

Example Show that an onto function f : {1, 2, 3} → {1, 2, 3} is always one-one.

Solution Suppose f is not one-one. Then there exists two elements, say 1 and 2 in the domain whose image in the co-domain is same. Also, the image of 3 under f can be only one element. Therefore, the range set can have at the most two elements of the co-domain {1, 2, 3}, showing that f is not onto, a contradiction. Hence, f must be one-one.

Example Show that a one-one function f : {1, 2, 3} → {1, 2, 3} must be onto.

Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different elements of the co-domain {1, 2, 3} under f. Hence, f has to be onto.

# IIT JEE Maths Study Material

Maths |

Algebra |

Sets |

Sequences and series |

Complex Numbers |

Quadratic Equations |

Linear inequalities |

Permutations and Combinations |

Probability |

Binomial theorem |

Matrices and determinants |

Trignometry |

Calculus |

Relations and Functions |

Limits and Derivatives |

Continuity and Differentiability |

Applications of derivatives |

Integral calculus |

Application of integrals |

Differential equations |

Coordinate Geometry |

Vector and 3D Geometry |