Give an example of a function which is
(i) one - one but not onto
(ii) one - one and onto
(iii) neither one - one nor onto
(iv) onto but not one - one.
(i) one - one but not onto
$f(x)=6 x$
For One - One
$f\left(x_{1}\right)=6 x_{1}$
$f\left(x_{2}\right)=6 x_{2}$
put $f\left(x_{1}\right)=f\left(x_{2}\right)$ we get
$6 x_{1}=6 x_{2}$
Hence, if $f\left(x_{1}\right)=f\left(x_{2}\right), x_{1}=x_{2}$
Function $f$ is one - one
For Onto
$f(x)=6 x$
let $f(x)=y$, such that $y \in N$
$6 x=y$
$\Rightarrow x=\frac{y}{6}$
If $y=1$
$x=\frac{1}{6}=0.166667$
which is not possible as $x \in N$
Hence, $f$ is not onto.
(ii) one - one and onto
$f(x)=x^{5}$
$\Rightarrow y=x^{5}$
Since the lines do not cut the curve in 2 equal valued points of $y$, therefore, the function $f(x)$ is one - one.
The range of $f(x)=(-\infty, \infty)=R$ (Codomain)
$\therefore f(x)$ is onto
$\therefore f(x)$ is one - one and onto.
(iii) neither one - one nor onto
$f(x)=x^{2}$
for one one:
$f\left(x_{1}\right)=\left(x_{1}\right)^{2}$
$f\left(x_{2}\right)=\left(x_{2}\right)^{2}$
$f\left(x_{1}\right)=f\left(x_{2}\right)$
$\Rightarrow\left(x_{1}\right)^{2}=\left(x_{2}\right)^{2}$
$\Rightarrow x_{1}=x_{2}$ or $x_{1}=-x_{2}$
Since $x_{1}$ does not have a unique image it is not one - one
For onto
$f(x)=y$
such that $y \in R$
$x^{2}=y$
$\Rightarrow x=\pm \sqrt{y}$
If $y$ is negative under root of a negative number is not real
Hence, $f(x)$ is not onto.
$\therefore f(x)$ is neither onto nor one - one
(iv) onto but not one - one.
Consider a function $f: Z \rightarrow N$ such that $f(x)=|x| .$
Since the Z maps to every single element in N twice, this function is onto but not one - one.
$Z$ - integers
N - natural numbers.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,