Author: Rishabh Kumar (IIT + ISI | Global Math Mentor)
Published: October 2025
Category: Algebra | Functions & Graph Analysis
🔹 Why This Matters
Every function tells a story — but to understand it fully, we must know:
- Where it exists → Domain
- What values it can take → Range
- How it behaves near boundaries or infinity → Asymptotes
Together, these three concepts give the complete picture of any function’s behavior — algebraically and geometrically.
🧭 1️⃣ Domain of a Function
The domain is the set of all input values (x) for which the function is defined.
Simply put: the domain tells you which x-values work.
🔹 How to Find the Domain
- Check for division by zero
Denominator ≠ 0 Example: f(x)=1x−3⇒x≠3f(x) = \frac{1}{x – 3} \Rightarrow x \neq 3f(x)=x−31⇒x=3 ✅ Domain: R−{3}\mathbb{R} – \{3\}R−{3}
- Check for square roots or even roots
The inside must be ≥ 0 (for real-valued functions). Example: f(x)=2x−4⇒2x−4≥0⇒x≥2f(x) = \sqrt{2x – 4} \Rightarrow 2x – 4 \ge 0 \Rightarrow x \ge 2f(x)=2x−4⇒2x−4≥0⇒x≥2 ✅ Domain: [2,∞)[2, \infty)[2,∞)
- Check for logarithms
The argument must be > 0. Example: f(x)=ln(x−1)⇒x−1>0⇒x>1f(x) = \ln(x – 1) \Rightarrow x – 1 > 0 \Rightarrow x > 1f(x)=ln(x−1)⇒x−1>0⇒x>1 ✅ Domain: (1,∞)(1, \infty)(1,∞)
- Check for composite or nested functions
Solve inner restrictions before outer ones. Example: f(x)=x−2x+1f(x) = \sqrt{\frac{x – 2}{x + 1}} f(x)=x+1x−2 x−2x+1≥0, x≠−1\frac{x – 2}{x + 1} \ge 0, \ x \neq -1x+1x−2≥0, x=−1 ✅ Domain: (−∞,−1)∪[2,∞)(-\infty, -1) \cup [2, \infty)(−∞,−1)∪[2,∞)
🎯 2️⃣ Range of a Function
The range is the set of all output values (y) the function can produce.
The range tells you how far the function can go vertically.
🔹 How to Find the Range
- Direct substitution or reasoning
If f(x)=x2f(x) = x^2f(x)=x2, since x2≥0x^2 ≥ 0x2≥0,
✅ Range: [0,∞)[0, \infty)[0,∞) - Express x in terms of y
Example: y=x+2⇒x=y2−2y = \sqrt{x + 2} \Rightarrow x = y^2 – 2y=x+2⇒x=y2−2 Since the square root is ≥ 0,
✅ Range: [0,∞)[0, \infty)[0,∞) - Use limits for end behavior
For f(x)=xx+1f(x) = \frac{x}{x+1}f(x)=x+1x:
As x→∞,f(x)→1x → ∞, f(x) → 1x→∞,f(x)→1; as x→−∞,f(x)→1x → -∞, f(x) → 1x→−∞,f(x)→1 from below.
✅ Range: (−∞,1)∪(1,∞)(-∞, 1) \cup (1, ∞)(−∞,1)∪(1,∞) or (−∞,1)(-∞, 1)(−∞,1) if function restricted.
🧩 Example — f(x)=x2x2+1f(x) = \frac{x^2}{x^2 + 1}f(x)=x2+1×2
- Denominator never zero → domain = R\mathbb{R}R
- Since 0≤x2x2+1<10 ≤ \frac{x^2}{x^2 + 1} < 10≤x2+1×2<1
✅ Range: [0,1)[0, 1)[0,1)
🧮 3️⃣ Asymptotes of a Function
An asymptote is a line that the graph approaches but never touches as xxx or yyy tends to infinity or a critical point.
They reveal a function’s long-term behavior and boundaries.
🔹 Types of Asymptotes
| Type | Equation | Meaning |
|---|---|---|
| Vertical | x=ax = ax=a | Graph blows up (denominator → 0) |
| Horizontal | y=by = by=b | Graph approaches constant value as x→±∞x → ±∞x→±∞ |
| Oblique (Slant) | y=mx+cy = mx + cy=mx+c | Graph follows a line for large |
🔹 Vertical Asymptotes
Occurs when the denominator = 0 (but numerator ≠ 0).
🧩 Example: f(x)=1x−2f(x) = \frac{1}{x – 2}f(x)=x−21
At x=2x = 2x=2, denominator = 0 →
✅ Vertical asymptote: x=2x = 2x=2
🔹 Horizontal Asymptotes
Found using limit as x→∞x → ∞x→∞ or degree comparison.
- If degrees of numerator and denominator are equal: y=leading coefficient of numeratorleading coefficient of denominatory = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}y=leading coefficient of denominatorleading coefficient of numerator
🧩 Example: f(x)=2×2+3×2+1⇒y=2f(x) = \frac{2x^2 + 3}{x^2 + 1} \Rightarrow y = 2f(x)=x2+12×2+3⇒y=2
✅ Horizontal asymptote: y=2y = 2y=2
🔹 Oblique (Slant) Asymptote
Occurs when degree of numerator = degree of denominator + 1.
Use polynomial long division.
🧩 Example: f(x)=x2+1x−1f(x) = \frac{x^2 + 1}{x – 1}f(x)=x−1×2+1
Divide: x2+1=(x−1)(x+1)+2x^2 + 1 = (x – 1)(x + 1) + 2×2+1=(x−1)(x+1)+2
So f(x)=x+1+2x−1f(x) = x + 1 + \frac{2}{x – 1}f(x)=x+1+x−12
✅ Oblique asymptote: y=x+1y = x + 1y=x+1
🔹 Summary Table
| Function | Domain | Range | Asymptotes |
|---|---|---|---|
| 1x\frac{1}{x}x1 | x≠0x ≠ 0x=0 | y≠0y ≠ 0y=0 | Vertical: x=0x=0x=0, Horizontal: y=0y=0y=0 |
| exe^xex | R\mathbb{R}R | (0,∞)(0, ∞)(0,∞) | Horizontal: y=0y=0y=0 |
| tanx\tan xtanx | x≠π2+nπx ≠ \frac{\pi}{2} + n\pix=2π+nπ | R\mathbb{R}R | Vertical: x=π2+nπx=\frac{\pi}{2}+n\pix=2π+nπ |
| lnx\ln xlnx | (0,∞)(0, ∞)(0,∞) | R\mathbb{R}R | Vertical: x=0x=0x=0 |
🔹 Common Mistakes
- ❌ Forgetting to exclude zero denominators from domain.
- ❌ Mixing up horizontal vs oblique asymptotes.
- ❌ Assuming range = domain.
- ❌ Not checking for even roots or log restrictions.
🌟 Why This Matters
Understanding domain, range, and asymptotes makes you fluent in function behavior — critical for:
- Sketching graphs precisely
- Solving inequalities
- Analyzing limits and continuity
- Preparing for advanced calculus or physics
This topic bridges algebra and calculus — structure meets motion.
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