Show that the function $f: R \rightarrow R: f(x)=1+x^{2}$ is many-one into.
To prove: function is many-one into
Given: $f: R \rightarrow R: f(x)=1+x^{2}$
We have,
$f(x)=1+x^{2}$
For, $f\left(x_{1}\right)=f\left(x_{2}\right)$
$\Rightarrow 1+x_{1}^{2}=1+x_{2}^{2}$
$\Rightarrow x_{1}^{2}=x_{2}^{2}$
$\Rightarrow x_{1}^{2}-x_{2}^{2}=0$
$\Rightarrow\left(x_{1}-x_{2}\right)\left(x_{1}+x_{2}\right)=0$
$\Rightarrow x_{1}=x_{2}$ or, $x_{1}=-x_{2}$
Clearly $x_{1}$ has more than one image
$\therefore f(x)$ is many-one
$f(x)=1+x^{2}$
Let $f(x)=y$ such that $y \in R$
$\Rightarrow y=1+x^{2}$
$\Rightarrow x^{2}=y-1$
$\Rightarrow x=\sqrt{y-1}$
If $y=3$, as $y \in R$
Then $x$ will be undefined as we can't place the negative value under the square root
Hence $f(x)$ is into
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,