Question:
Let $R_{0}$ be the set of all nonzero real numbers. Then, show that the function $f: R_{0} \rightarrow R_{0}: f(x)=\frac{1}{x}$ is one-
one and onto.
Solution:
To prove: function is one-one and onto
Given: $f: R_{0} \rightarrow R_{0}: f(x)=\frac{1}{x}$
We have,
$f(x)=\frac{1}{x}$
For, $f\left(x_{1}\right)=f\left(x_{2}\right)$
$\Rightarrow \frac{1}{x_{1}}=\frac{1}{x_{2}}$
$\Rightarrow x_{1}=x_{2}$
When, $f\left(x_{1}\right)=f\left(x_{2}\right)$ then $x_{1}=x_{2}$
$\Rightarrow y=\frac{1}{x}$
$\Rightarrow x=\frac{1}{y}$
Since $y \in R_{0}$,
$\Rightarrow x$ will also $\in R_{0}$, which means that every value of $y$ is associated with some $x$
$\therefore f(x)$ is onto
Hence Proved
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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,
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