For $\mathrm{x} \in \mathrm{R}-\{0,1\}$, let $\mathrm{f}_{1}(\mathrm{x})=\frac{1}{\mathrm{x}}$
$\mathrm{f}_{2}(\mathrm{x})=1-\mathrm{x}$ and $\mathrm{f}_{3}(\mathrm{x})=\frac{1}{1-\mathrm{x}}$ be three
given functions. If a function, $\mathrm{J}(\mathrm{x})$ satisfies $\left(f_{2}{ }^{\circ}{ }^{\circ} \mathrm{f}_{1}\right)(\mathrm{x})=\mathrm{f}_{3}(\mathrm{x})$ then $\mathrm{J}(\mathrm{x})$ is equal to :-
$\mathrm{f}_{3}(\mathrm{x})$
$f_{1}(x)$
$\mathrm{f}_{2}(\mathrm{x})$
$\frac{1}{x} f_{3}(x)$
Correct Option: 1
Given $\mathrm{f}_{1}(\mathrm{x})=\frac{1}{\mathrm{x}}, \mathrm{f}_{2}(\mathrm{x})=1-\mathrm{x}$ and $\mathrm{f}_{3}(\mathrm{x})=\frac{1}{1-\mathrm{x}}$
$\left(\mathrm{f}_{2} \circ \mathrm{Jof} \mathrm{f}_{1}\right)(\mathrm{x})=\mathrm{f}_{3}(\mathrm{x})$
$\mathrm{f}_{2} \circ\left(\mathrm{J}\left(\mathrm{f}_{1}(\mathrm{x})\right)\right)=\mathrm{f}_{3}(\mathrm{x})$
$f_{2} \circ\left(J\left(\frac{1}{x}\right)\right)=\frac{1}{1-x}$
$1-J\left(\frac{1}{x}\right)=\frac{1}{1-x}$
$J\left(\frac{1}{x}\right)=1-\frac{1}{1-x}=\frac{-x}{1-x}=\frac{x}{x-1}$
Now $x \rightarrow \frac{1}{x}$
$J(x)=\frac{\frac{1}{x}}{\frac{1}{x}-1}=\frac{1}{1-x}=f_{3}(x)$
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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,