Let $f:\left[0, \frac{\pi}{2}\right] \rightarrow R: f(x)=\sin x$ and $g:\left[0, \frac{\pi}{2}\right] \rightarrow R: g(x)=\cos x$ Show that each one of $f$ and $g$ is one - one but $(f+g)$ is not one - one.
$f:\left[0, \frac{\pi}{2}\right] \rightarrow R: f(x)=\sin x$
Here in this range, the lines do not cut the curve in 2 equal valued points of $y$, therefore, the function $f(x)=$ $\sin x$ is one - one.
$g:\left[0, \frac{\pi}{2}\right] \rightarrow R: g(x)=\cos x$
in this range, the lines do not cut the curve in 2 equal valued points of $y$, therefore, the function $f(x)=$ cosx is also one - one.
$(f+g):\left[0, \frac{\pi}{2}\right] \rightarrow R=\sin x+\cos x$
in this range the lines cut the curve in 2 equal valued points of $y$, therefore, the function $f(x)=$ cosx $+$ sin $x$ is not one - one.
Hence,showed that each one of $f$ and $g$ is one - one but $(f+g)$ is not one - one.
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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,