Question:
Prove that the function $f: R \rightarrow R: f(x)=2 x$ is one-one and onto.
Solution:
To prove: function is one-one and onto
Given: $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}: \mathrm{f}(\mathrm{x})=2 \mathrm{x}$
We have,
$f(x)=2 x$
For, $f\left(x_{1}\right)=f\left(x_{2}\right)$
$\Rightarrow 2 x_{1}=2 x_{2}$
$\Rightarrow x_{1}=x_{2}$
When, $f\left(x_{1}\right)=f\left(x_{2}\right)$ then $x_{1}=x_{2}$
∴ f(x) is one-one
$f(x)=2 x$
Let $f(x)=y$ such that $y \in R$
$\Rightarrow y=2 x$
$\Rightarrow x=\frac{y}{2}$
Since $y \in R$,
$\Rightarrow \frac{y}{2} \in R$
⇒ x will also be a real number, 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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