Find the domain of each of the following functions given by
(i) $f(x)=\frac{1}{\sqrt{1-\cos x}}$
(ii) $f(x)=\frac{1}{\sqrt{x+|x|}}$
(iii) $f(x)=x|x|$
(iv) $f(x)=\frac{x^{3}-x+3}{x^{2}-1}$
(v) $f(x)=\frac{3 x}{2 x-8}$
(i)
$f(x)=\frac{1}{\sqrt{1-\cos x}}$
According to the question,
We know the value of cos x lies between –1, 1,
–1 ≤ cos x ≤ 1
Multiplying by negative sign, we get
Or 1 ≥ – cos x ≥ –1
Adding 1, we get
2 ≥ 1– cos x ≥ 0 …(i)
Now,
$f(x)=\frac{1}{\sqrt{1-\cos x}}$
1– cos x ≠ 0
⇒ cos x ≠ 1
Or, x ≠ 2nπ ∀ n ∈ Z
Therefore, the domain of f = R–{2nπ:n∈Z}
(ii)
$f(x)=\frac{1}{\sqrt{x+|x|}}$
According to the question,
For real value of f,
x + |x| > 0
When x > 0,
x + |x| > 0⇒ x + x > 0 ⇒ 2x > 0⇒ x > 0
When x < 0,
x + |x| > 0⇒ x – x > 0 ⇒ 2x > 0⇒ x > 0
So, x > 0, to satisfy the given equation.
Therefore, the domain of f = R+
(iii)
f(x) = x|x|
According to the question,
We know x and |x| are defined for all real values.
Therefore, the domain of f = R
(iv)
$f(x)=\frac{\left(x^{3}-x+3\right)}{x^{2}-1}$
According to the question,
For real value of
x2–1≠0
⇒ (x–1)(x + 1)≠0
⇒ x–1≠0 or x + 1≠0
⇒ x≠1 or x≠–1
Therefore, the domain of f = R–{–1, 1}
(v)
$f(x)=\frac{3 x}{2 x-8}$
According to the question,
For real value of
28 – x ≠0
⇒ x≠ 28
Therefore, the domain of f = R–{28}
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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,