Let $f(\mathrm{x})=\mathrm{a}^{\mathrm{x}}(\mathrm{a}>0)$ be written as $f(\mathrm{x})=f_{1}(\mathrm{x})+f_{2}(\mathrm{x})$, where $f_{1}(\mathrm{x})$ is an even function of $f_{2}(\mathrm{x})$ is an odd function. Then $f_{1}(\mathrm{x}+\mathrm{y})+f_{1}(\mathrm{x}-\mathrm{y})$ equals
$2 f_{1}(\mathrm{x}) f_{1}(\mathrm{y})$
$2 f_{1}(\mathrm{x}) f_{2}(\mathrm{y})$
$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{2}(\mathrm{x}-\mathrm{y})$
$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{1}(\mathrm{x}-\mathrm{y})$
Correct Option: 1
$f(x)=a^{x}, a>0$
$f(x)=\frac{a^{x}+a^{-x}+a^{x}-a^{-x}}{2}$
$\Rightarrow f_{1}(x)=\frac{a^{x}+a^{-x}}{2}$
$f_{2}(\mathrm{x})=\frac{\mathrm{a}^{\mathrm{x}}-\mathrm{a}^{-\mathrm{x}}}{2}$
$\Rightarrow f_{1}(\mathrm{x}+\mathrm{y})+f_{1}(\mathrm{x}-\mathrm{y})$
$=\frac{a^{x+y}+a^{-x-y}}{2}+\frac{a^{x-y}+a^{-x+y}}{2}$
$=\frac{\left(a^{x}+a^{-x}\right)}{2}\left(a^{y}+a^{-y}\right)$
$=f_{1}(\mathrm{x}) \times 2 f_{1}(\mathrm{y})$
$=2 f_{1}(\mathrm{x}) f_{1}(\mathrm{y})$
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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,