Vidaara.orgClass 12 · Mathematics
CodeVID-M12-01-FUN-01
Functions: One-One, Onto & Bijective — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
The function $f:\mathbb{R}\to\mathbb{R},\ f(x)=x^3$ is:
- A.one-one but not onto
- B.onto but not one-one
- C.bijective
- D.neither
2.
$f:\mathbb{R}\to\mathbb{R},\ f(x)=x^2$ is:
- A.one-one
- B.onto
- C.bijective
- D.neither one-one nor onto
3.
The number of one-one functions from a set of $3$ elements to a set of $4$ elements is:
- A.$12$
- B.$24$
- C.$64$
- D.$81$
4.
$f:\mathbb{R}\to\mathbb{R},\ f(x)=3x-7$ is:
- A.many-one
- B.bijective
- C.not a function
- D.onto but not one-one
5.
$f:\mathbb{N}\to\mathbb{N},\ f(x)=x^2$ is:
- A.one-one but not onto
- B.onto but not one-one
- C.bijective
- D.neither
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Show that $f:\mathbb{R}\to\mathbb{R},\ f(x)=2x+5$ is one-one.
7.
Is $f:\mathbb{R}\to\mathbb{R},\ f(x)=|x|$ onto? Justify.
8.
Give an example of a function $\mathbb{R}\to\mathbb{R}$ that is onto but not one-one.
9.
If $f:\{1,2,3\}\to\{a,b,c\}$ is one-one, is it necessarily onto?
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Show that $f:\mathbb{R}\to\mathbb{R},\ f(x)=2x+3$ is a bijection.
11.
Examine $f:\mathbb{R}\to\mathbb{R},\ f(x)=x^2$ for injectivity and surjectivity.
12.
Prove that $f:\mathbb{N}\to\mathbb{N},\ f(x)=x^2$ is one-one but not onto.
13.
Is $f:\mathbb{R}\setminus\{3\}\to\mathbb{R},\ f(x)=\dfrac{x-2}{x-3}$ one-one?
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Show that $f:\mathbb{R}\to\mathbb{R},\ f(x)=x^2$ is neither one-one nor onto, and give a restriction of domain and codomain that makes it bijective.
15.
Let $A=\mathbb{R}\setminus\{3\}$ and $B=\mathbb{R}\setminus\{1\}$. Show that $f:A\to B,\ f(x)=\dfrac{x-2}{x-3}$ is both one-one and onto.
Answer Key
Section A — Multiple Choice Questions
- (C) bijective
- (D) neither one-one nor onto
- (B) $24$
- (B) bijective
- (A) one-one but not onto
Section B — Short Answer (2 marks)
- One-one.
- No, not onto (range $=[0,\infty)$).
- e.g. $f(x)=x^3-x$.
- Yes (equal finite sets).
Section C — Short Answer (3 marks)
- Bijective.
- Neither one-one nor onto.
- One-one but not onto.
- Yes, one-one.
Section D — Long Answer (5 marks)
- Neither on $\mathbb{R}$; bijective as $f:[0,\infty)\to[0,\infty)$.
- $f$ is a bijection from $A$ to $B$.
Generated by Vidaara.org · Assignment VID-M12-01-FUN-01 · vidaara.org