Domain & Range: The Foundation of JEE Calculus
Complete guide to mastering domain and range concepts for JEE Main & Advanced with step-by-step methods and solved examples.
Why Domain & Range Matter for JEE
Domain and Range form the bedrock of calculus and are among the most frequently tested concepts in JEE Main and Advanced. Understanding these concepts is crucial because:
- They appear directly in 2-3 questions every year
- They're essential for understanding limits, continuity, and differentiability
- Mistakes in domain/range lead to errors in integration and differentiation
- They help visualize function behavior graphically
What are Domain and Range?
Domain
The set of all possible input values (x-values) for which the function is defined.
Range
The set of all possible output values (y-values) that the function can produce.
Example 1: Basic Understanding
For $f(x) = x^2$:
- Domain: All real numbers $\mathbb{R}$ (we can square any real number)
- Range: $[0, \infty)$ (squares are always non-negative)
The 4 Golden Rules for Finding Domain
1. Denominator Cannot Be Zero
For $f(x) = \frac{g(x)}{h(x)}$, we require $h(x) \neq 0$
Example: $f(x) = \frac{1}{x-2}$
Domain: $x - 2 \neq 0 \Rightarrow x \neq 2$
$\therefore$ Domain = $\mathbb{R} - \{2\}$
2. Expression Inside Square Root (Even Root) Must Be ≥ 0
For $f(x) = \sqrt{g(x)}$, we require $g(x) \geq 0$
Example: $f(x) = \sqrt{x-3}$
Domain: $x - 3 \geq 0 \Rightarrow x \geq 3$
$\therefore$ Domain = $[3, \infty)$
3. Expression Inside Logarithm Must Be > 0
For $f(x) = \log(g(x))$, we require $g(x) > 0$
Example: $f(x) = \ln(x+4)$
Domain: $x + 4 > 0 \Rightarrow x > -4$
$\therefore$ Domain = $(-4, \infty)$
4. Special Restrictions for Inverse Trigonometric Functions
- $\sin^{-1}x, \cos^{-1}x$: Domain = $[-1, 1]$
- $\tan^{-1}x, \cot^{-1}x$: Domain = $\mathbb{R}$
- $\sec^{-1}x, \cosec^{-1}x$: Domain = $(-\infty, -1] \cup [1, \infty)$
Domain & Range of Algebraic Functions
| Function Type | Domain | Range |
|---|---|---|
| Polynomial $ax^n + ...$ | $\mathbb{R}$ | $\mathbb{R}$ (if odd degree) $[k, \infty)$ or $(-\infty, k]$ (if even degree) |
| Rational $\frac{p(x)}{q(x)}$ | $\mathbb{R} - \{\text{roots of } q(x)\}$ | Depends on function |
| Square Root $\sqrt{ax+b}$ | $[-\frac{b}{a}, \infty)$ if $a>0$ | $[0, \infty)$ |
Example 2: Complex Algebraic Function
Find domain of $f(x) = \frac{\sqrt{x-2}}{x^2-9}$
Step 1: Square root condition: $x - 2 \geq 0 \Rightarrow x \geq 2$
Step 2: Denominator condition: $x^2 - 9 \neq 0 \Rightarrow x \neq \pm 3$
Step 3: Combine: $x \geq 2$ AND $x \neq 3$
Solution: Domain = $[2, 3) \cup (3, \infty)$
Trigonometric Functions Domain & Range
Standard Trigonometric Functions
| Function | Domain | Range |
|---|---|---|
| $\sin x$ | $\mathbb{R}$ | $[-1, 1]$ |
| $\cos x$ | $\mathbb{R}$ | $[-1, 1]$ |
| $\tan x$ | $\mathbb{R} - \{\frac{\pi}{2} + n\pi\}$ | $\mathbb{R}$ |
Inverse Trigonometric Functions
| Function | Domain | Range |
|---|---|---|
| $\sin^{-1} x$ | $[-1, 1]$ | $[-\frac{\pi}{2}, \frac{\pi}{2}]$ |
| $\cos^{-1} x$ | $[-1, 1]$ | $[0, \pi]$ |
| $\tan^{-1} x$ | $\mathbb{R}$ | $(-\frac{\pi}{2}, \frac{\pi}{2})$ |
JEE Practice Problems
Problem 1: JEE Main 2023
The domain of the function $f(x) = \frac{1}{\sqrt{|x| - x}}$ is:
Solution Approach:
1. Denominator condition: $\sqrt{|x| - x} \neq 0 \Rightarrow |x| - x > 0$
2. Analyze $|x| - x > 0$:
• For $x \geq 0$: $|x| - x = x - x = 0$ ✗
• For $x < 0$: $|x| - x = -x - x = -2x > 0$ ✓
3. Domain: $x < 0$ or $(-\infty, 0)$
Problem 2: JEE Advanced 2022
Find the range of $f(x) = \frac{x^2 - 3x + 2}{x^2 + x - 6}$
Solution Approach:
1. Factor: $f(x) = \frac{(x-1)(x-2)}{(x-2)(x+3)} = \frac{x-1}{x+3}$ for $x \neq 2$
2. Let $y = \frac{x-1}{x+3}$ and solve for $x$: $x = \frac{1+3y}{1-y}$
3. Domain of this relation: $y \neq 1$
4. Also exclude $x=2$ case: when $x=2$, original function undefined
5. Range: $\mathbb{R} - \{1, \frac{1}{5}\}$
Quick Revision Checklist
Always Check For:
- Denominator ≠ 0
- Expression under even root ≥ 0
- Expression inside log > 0
- Special trig function restrictions
Common Mistakes:
- Forgetting to exclude points where function undefined
- Not considering all restrictions in composite functions
- Confusing domain of f(x) with f(g(x))
- Missing edge cases in piecewise functions
Ready for the Next Topic?
Mastered Domain & Range? Continue your JEE Calculus journey with:
Limits & Continuity →