Show that the function
(i) $f: N \rightarrow N: f(x)=x^{2}$ is one - one into.
(ii) $f: Z \rightarrow Z: f(x)=x^{2}$ is many - one into
(i) $f: N \rightarrow N: f(x)=x^{2}$ is one - one into.
$f(x)=x^{2}$
$\Rightarrow y=x^{2}$
Since the function $f(x)$ is monotonically increasing from the domain $N \rightarrow N$
$\therefore f(x)$ is one -one
Range of $f(x)=(0, \infty) \neq N($ codomain $)$
$\therefore f(x)$ is into
$\therefore f: N \rightarrow N: f(x)=x^{2}$ is one - one into.
(ii) $f: Z \rightarrow Z: f(x)=x^{2}$ is many - one into
$f(x)=x^{2}$
$\Rightarrow y=x^{2}$
in this range the lines cut the curve in 2 equal valued points of $y$, therefore, the function $f(x)=x^{2}$ is many one.
Range of $f(x)=(0, \infty) \neq Z$ (codomain)
$\therefore f(x)$ is into
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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,