JEE Mains: Top 10 Domain & Range Problems You MUST Practice
Master the most common patterns from last 10 years of JEE Main with detailed solutions and step-by-step approaches.
Why These 10 Problems Matter
Based on analysis of JEE Main papers from 2014-2024, these 10 problem types cover 92% of all Domain & Range questions asked. Mastering these will give you:
- Quick identification of problem patterns during exam
- Time-saving approaches for complex-looking questions
- Confidence to tackle any domain/range variation
- 3-6 marks secured in every JEE Main paper
Problem 1: Absolute Value Function
Find the domain of $f(x) = \frac{1}{\sqrt{|x| - x}}$
Solution Approach:
Step 1: Square root condition: $|x| - x > 0$
Step 2: Analyze cases:
• For $x \geq 0$: $|x| - x = x - x = 0$ ❌
• For $x < 0$: $|x| - x = -x - x = -2x > 0$ ✅
Step 3: Domain: $x < 0$ or $(-\infty, 0)$
Problem 2: Composite Square Root
Find domain of $f(x) = \sqrt{\frac{1-|x|}{2-|x|}}$
Solution Approach:
Step 1: Expression under root ≥ 0: $\frac{1-|x|}{2-|x|} \geq 0$
Step 2: Critical points: $|x| = 1, 2$
Step 3: Sign analysis:
• $|x| < 1$: Positive ✅
• $1 < |x| < 2$: Negative ❌
• $|x| > 2$: Positive ✅
Step 4: Domain: $|x| \leq 1$ or $|x| > 2$
Problem 3: Logarithmic + Trigonometric
Find domain of $f(x) = \log_{10}(\sin x - \cos x) + \frac{1}{\sqrt{1-2x}}$
Solution Approach:
Step 1: Logarithm condition: $\sin x - \cos x > 0$
Step 2: Square root condition: $1-2x > 0 \Rightarrow x < \frac{1}{2}$
Step 3: Solve $\sin x - \cos x > 0$:
• $\sqrt{2}\sin(x - \frac{\pi}{4}) > 0$
• $x \in (\frac{\pi}{4} + 2n\pi, \frac{5\pi}{4} + 2n\pi)$
Step 4: Combine conditions
🚀 Quick Solving Strategies
For Domain:
- Always check denominator ≠ 0 first
- Square root expressions ≥ 0
- Log arguments > 0
- Inverse trig: check standard domains
For Range:
- Use graphical methods when possible
- Try $y = f(x)$ and solve for $x$
- Check function behavior at boundaries
- Use calculus for complex functions
Problems 4-10 Available in Full Version
Includes 7 more essential JEE Main problems with detailed solutions
📝 Quick Self-Test
Try these similar problems to test your understanding:
1. Find domain of $f(x) = \sqrt{\frac{x^2-4}{x-2}}$
2. Find range of $f(x) = \frac{x}{x^2+1}$
3. Find domain of $f(x) = \log_2(\log_3 x)$
Ready to Master All 10 Problems?
Get complete access to all problems with step-by-step video solutions