If $t, g$ and $h$ are real functions defined by $f(x)=\sqrt{x+1}, g(x)=\frac{1}{x}$ and $h(x)=2 x^{2}-3$, find the values of $(2 f+g-h)(1)$ and $(2 f+g-h)(0)$.
Given:
$f(x)=\sqrt{x+1}, g(x)=\frac{1}{x}$ and $h(\mathrm{x})=2 \mathrm{x}^{3}-3$
Clearly, f (x) is defined for x + 1 ≥ 0 .
$\Rightarrow x \geq-1$
$\Rightarrow x \in[-1, \infty]$
Thus, domain ( f ) = [
Clearly, g (x) is defined for x ≠ 0 .
⇒ x ∈ R – { 0} and h(x) is defined for all x such that x ∈ R .
Thus,
domain ( f ) ∩ domain (g) ∩ domain (h) = [
Hence,
$(2 f+g-h):[-1, \infty]-\{0\} \rightarrow \mathrm{R}$ is given by:
$(2 f+g-h)(x)=2 f(x)+g(x)-h(x)$
$=2 \sqrt{x+1}+\frac{1}{x}-2 x^{2}+3$
$(2 f+g-h)(1)=2 \sqrt{2}+1-2+3=2 \sqrt{2}+4-2=2 \sqrt{2}+2$
$(2 f+g-h)(0)$ does not exist because 0 does not lie in the domain $x \in[-1, \infty]-\{0\}$
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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,