Question:
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f (x) = x – 2/ x – 3 ∀ x ∈ A . Then show that f is bijective.
Solution:
Given,
A = R – {3}, B = R – {1}
And,
f : A → B be defined by f (x) = x – 2/ x – 3 ∀ x ∈ A
Hence, f (x) = (x – 3 + 1)/ (x – 3) = 1 + 1/ (x – 3)
Let f(x1) = f (x2)
$1+\frac{1}{x_{1}-3}=1+\frac{1}{x_{2}-3}$
$\frac{1}{x_{1}-3}=\frac{1}{x_{2}-3}$
$x_{1}=x_{2}$
So, f (x) is an injective function.
Now let y = (x – 2)/ (x -3)
x – 2 = xy – 3y
x(1 – y) = 2 – 3y
x = (3y – 2)/ (y – 1)
y ∈ R – {1} = B
Thus, f (x) is onto or subjective.
Therefore, f(x) is a bijective function.
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- 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
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- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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