Week 5 — functions & composition

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Week map

Classify → Represent → Execute → Trap-check

• Recognize: Ask: Which test checks if a graph is a function?
• Procedure: Write $f(g(x))$ by substituting entire $g(x)$ into $f$. Simplify stepwise; watch parentheses when replacing $x$. Domain excludes values that break inside function.
• Variations / traps: Watch for: Writing $f \circ g$ as $f(x) \cdot g(x)$.

Formula chain (compressed)

domain/range → (f∘g)(x)=f(g(x)) → inverse swaps x↔y → exp growth vs decay.

1. Composition — (f∘g)(x) = f(g(x)) — inside-out evaluation order
2. Inverse — f⁻¹(f(x))=x on domain — one-to-one functions only
3. Horizontal line test — no horizontal hits twice — invertible on interval
4. Exponential — f(x)=a^x, a>0, a≠1 — growth (a>1) vs decay (0<a<1)
5. Domain of composite — x ∈ dom(g) and g(x) ∈ dom(f) — before plugging in

Deep study

Mathematics I · Week 5 — Composition, inverse, and exponentials

Week map

Composition notation

• $(f \circ g)(x)$ → “$f$ composed with $g$ at $x$” → apply $g$ first, then $f$ → $(f \circ g)(x) = f(g(x))$.
• $f(g(x))$ → inner function output becomes outer input → parentheses matter when substituting.
• Domain of composition → all $x$ where $g(x)$ is in domain of $f$.

Vertical and horizontal line tests

• Vertical line test: graph is a function iff no vertical line hits twice.
• Horizontal line test: function is injective (one-to-one) iff no horizontal line hits twice — needed for invertible function on its domain.

Inverse notation

• $f^{-1}(y)$ → input that $f$ sends to $y$ → not the same as $1/f(x)$.
• Inverse swaps roles: if $f(a) = b$ then $f^{-1}(b) = a$.
• Domain of $f^{-1}$ = range of $f$; range of $f^{-1}$ = domain of $f$.
• Graph of $f^{-1}$ reflects graph of $f$ over line $y = x$.

Exponential notation

• $y = a^x$ → exponential with base $a$ → $a > 0$, $a \neq 1$.
• $a^0 = 1$ for any valid base; $a^{m+n} = a^m \cdot a^n$; $(a^m)^n = a^{mn}$.
• $a^x > 0$ always — range is $(0, \infty)$.

Pattern families

Easy — Evaluate composition

Medium — Find inverse of linear or simple rational

Hard — Invertibility and restricted domains

Worked mini-examples

Traps

• Writing $(f \circ g)(x)$ as $f(x) \cdot g(x)$.
• Treating $f^{-1}$ as $1/f(x)$.
• Taking domain of $f^{-1}$ equal to domain of $f$.
• Applying outer function before inner in composition.
• Confusing exponential base with exponent: $2^x$ vs $x^2$.

Diagnostic (try yourself)

1. If $f(x) = 3x - 1$ and $g(x) = x^2$, find $(g \circ f)(2)$ and $(f \circ g)(2)$.
2. Find $f^{-1}(x)$ for $f(x) = 5x + 4$. What is the domain of $f^{-1}$ if domain of $f$ is $\mathbb{R}$?
3. Is $f(x) = |x|$ injective on $\mathbb{R}$? What restriction makes it bijective from $\mathbb{R}$ to $[0,\infty)$?
4. Simplify $(f \circ g)(x)$ when $f(x) = \sqrt{x}$ and $g(x) = x + 5$. State the domain of the composition.
5. Evaluate $2^{3}$ and $3^{2}$. Which is larger? What is $2^{-3}$?

ChatGPT prep archive

• Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

• Vertical line test: function iff no vertical line hits graph twice.
• Horizontal line test: injective (one-to-one) iff no horizontal line hits twice—needed for invertible function.
• Composition $(f \circ g)(x) = f(g(x))$: inside function first.
• Inverse $f^{-1}$ swaps domain/range; graph reflection over $y=x$. Exponential $a^x$, $a>0$, $a \neq 1$.

Notation & vocabulary

Pattern families

Easy — Compose two formulas

Medium — Inverse of linear or simple rational

Hard — One-to-one and invertible

Drill these on the pattern atlas — filter to week 5.

Traps

• Writing $f \circ g$ as $f(x) \cdot g(x)$.
• Inverse notation $f^{-1}$ as $1/f$.
• Domain of $f^{-1}$ taken as domain of $f$.
• Base of exponential confused with exponent in $x^a$.

Retrieval prompts

1. Which test checks if a graph is a function?
2. In $f(g(x))$, which function is applied first?
3. What is the domain of $f^{-1}$ if $f$ maps $[1,5]$ to $[2,10]$?

Practice loop

1. Read Deep study (if present) or core concepts once.
2. Recite the formula chain without looking.
3. Open one easy pattern on the interactive atlas for week 5.
4. Attempt without solutions; mark studied after an honest try.
5. Say one trap aloud before closing the tab.

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk