Question:
Mark (√) against the correct answer in the following:
Let $f(x)=\frac{x^{2}}{\left(1+x^{2}\right)} .$ Then, range $(f)=?$
A. $[1, \infty)$
B. $[0,1)$
C. $[-1,1]$
D. $(0,1]$
Solution:
$\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{2}}$
$\Rightarrow \mathrm{y}=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{2}}$
$\Rightarrow \mathrm{y}+\mathrm{yx}^{2}=\mathrm{x}^{2}$
$\Rightarrow \mathrm{y}=\mathrm{x}^{2}(1-\mathrm{y})$
$\Rightarrow x=\sqrt{\frac{y}{1-y}}$
$\frac{y}{1-y} \geq 0$
$\Rightarrow y \geq 0$
And
$1-y>0$
$\Rightarrow y<1$
Taking intersection we get
range $(f)=[0,1)$
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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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