Continuity of Composite Functions: fog(x) and gof(x) | JEE Tricks & Shortcuts
Master when composite functions are continuous with solved examples, essential theorems, and time-saving shortcuts for JEE.
Why Composite Function Continuity Matters in JEE
Composite function continuity appears in 2-3 questions every year in JEE Main and Advanced. Understanding these concepts can help you secure 4-12 marks easily with proper application of theorems and shortcuts.
- Direct application in multiple-choice questions
- Crucial for solving complex limit problems
- Foundation for advanced calculus concepts
- Time-saving shortcuts for quick verification
Continuity of Composite Functions Theorem
Theorem Statement:
If $g$ is continuous at $x = a$ and $f$ is continuous at $g(a)$, then the composite function $f \circ g$ is continuous at $x = a$.
In mathematical notation: If $\lim_{x \to a} g(x) = g(a)$ and $\lim_{y \to g(a)} f(y) = f(g(a))$, then $\lim_{x \to a} f(g(x)) = f(g(a))$.
💡 JEE Shortcut:
Quick Check Method: For $f(g(x))$ to be continuous at $x=a$, both conditions must be satisfied:
- $g(x)$ is continuous at $x=a$
- $f(x)$ is continuous at $x=g(a)$
When Both Functions are Continuous Everywhere
If $f$ and $g$ are continuous on their entire domains, then $f \circ g$ and $g \circ f$ are also continuous.
Example: Check continuity of $f(g(x))$ where $f(x) = \sin x$ and $g(x) = x^2 + 1$
Step 1: Check continuity of $g(x)$: $g(x) = x^2 + 1$ is polynomial ⇒ continuous everywhere ✅
Step 2: Check continuity of $f(x)$: $f(x) = \sin x$ is trigonometric ⇒ continuous everywhere ✅
Step 3: Apply theorem: Since both are continuous everywhere, $f(g(x)) = \sin(x^2 + 1)$ is continuous everywhere ✅
🚀 Quick Rule:
Continuous + Continuous = Continuous
If both functions are continuous on $\mathbb{R}$, their composition is always continuous on $\mathbb{R}$.
When Inner Function has Discontinuity
If $g$ is discontinuous at $x=a$, then $f \circ g$ is usually discontinuous at $x=a$, regardless of $f$.
Example: $f(x) = x^2$, $g(x) = \frac{1}{x-1}$. Check continuity of $f(g(x))$ at $x=1$
Step 1: Analyze $g(x)$: $g(x) = \frac{1}{x-1}$ is discontinuous at $x=1$ ❌
Step 2: Check $f(x)$: $f(x) = x^2$ is continuous everywhere ✅
Step 3: Since $g$ is discontinuous at $x=1$, $f(g(x)) = \left(\frac{1}{x-1}\right)^2$ is discontinuous at $x=1$ ❌
⚠️ Important Exception:
If $f$ is constant at the point of discontinuity of $g$, the composition might still be continuous!
Example: If $f(x) = 5$ (constant) and $g$ is discontinuous at $a$, then $f(g(x)) = 5$ is continuous everywhere.
When Outer Function has Discontinuity
If $f$ is discontinuous at $y = g(a)$, then $f \circ g$ is discontinuous at $x=a$, even if $g$ is continuous.
JEE Main 2021: $f(x) = \begin{cases} x+1 & \text{if } x \geq 0 \\ x-1 & \text{if } x < 0 \end{cases}$, $g(x) = x^2$. Check continuity of $f(g(x))$
Step 1: Analyze $g(x)$: $g(x) = x^2$ is continuous everywhere ✅
Step 2: Analyze $f(x)$: $f(x)$ is discontinuous at $x=0$ (jump discontinuity) ❌
Step 3: Find where $g(x) = 0$: $x^2 = 0 ⇒ x=0$
Step 4: Since $f$ is discontinuous at $g(0)=0$, $f(g(x))$ is discontinuous at $x=0$ ❌
🚀 JEE Problem-Solving Framework
For fog(x) continuity at x=a:
- Check if g is continuous at x=a
- Find g(a)
- Check if f is continuous at g(a)
- If both yes ⇒ fog is continuous
Common Pitfalls to Avoid:
- Assuming composition preserves continuity
- Not checking domain restrictions
- Missing removable discontinuities
- Forgetting constant function exception
Concepts 4-6 Available in Full Version
Includes piecewise functions, trigonometric composites, and advanced JEE problems with detailed solutions
📝 Quick Self-Test
Try these JEE-level problems to test your understanding:
1. $f(x) = \sqrt{x}$, $g(x) = x^2 - 4$. Is $f(g(x))$ continuous at $x=2$?
2. $f(x) = \frac{1}{x}$, $g(x) = \sin x$. Find points of discontinuity of $f(g(x))$
3. $f(x) = |x|$, $g(x) = x^3 - x$. Check continuity of $g(f(x))$ at $x=0$
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