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Mathematics I · Week 5 — Functions & composition

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Mathematics I · Week 5 — Functions & composition

1280 words

6 min read

2026-08-16

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Now · Week map

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.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Function tests → composition order → inverse swap → exponential growth/decay

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.

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

Open interactive formula desk · Week 5 tab.

Deep study

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

Deep study for Quiz 2 week 5. Master composition order first, then invertibility, then exponential growth rules.

Week map

Vertical/horizontal line tests → composition

(f \circ g)

→ inverse

f^{-1}

→ exponential

a^x

→ domain discipline throughout.

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 $f$ .

Mini-example:

f(x) = 2x + 1

g(x) = x^2

. Then

(f \circ g)(3) = f(g(3)) = f(9) = 19

. But

(g \circ f)(3) = g(f(3)) = g(7) = 49

— order changes the answer.

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 $y = x$ .

Mini-example:

f(x) = 3x - 2

\mathbb{R}

. Swap:

x = 3y - 2

, solve

y = \frac{x+2}{3}

. So

f^{-1}(x) = \frac{x+2}{3}

. Check:

f^{-1}(5) = 7/3

and

f(7/3) = 5

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 $(0, \infty)$ .

Mini-example:

2^3 = 8

;

2^{-1} = 1/2

;

(1/2)^2 = 1/4

. Base between 0 and 1 gives decay; base

> 1

gives growth.

Trap:

x^a

(power function) vs

a^x

(exponential) — variable position differs.

Pattern families

Easy — Evaluate composition

Substitute entire inner expression into outer. Simplify stepwise. Watch signs when replacing

x

inside

f

Medium — Find inverse of linear or simple rational

Swap

x

and

y

, solve for

y

. State domain of inverse explicitly. Verify

f(f^{-1}(x)) = x

on valid domain.

Hard — Invertibility and restricted domains

Use horizontal line test or monotonicity. Square root and squaring need domain restriction (

x \geq 0

) for bijection. Composition with inverse simplifies to identity only on matched domains.

Worked mini-examples

Example 1 — Composition order.

f(x) = x + 4

g(x) = 2x

(f \circ g)(5) = f(10) = 14

(g \circ f)(5) = g(9) = 18

Example 2 — Nested formula.

f(x) = x^2 + 1

g(x) = x - 3

(f \circ g)(x) = (x-3)^2 + 1 = x^2 - 6x + 10

Example 3 — Inverse of linear.

f(x) = \frac{x-1}{2}

. Swap:

x = \frac{y-1}{2}

→

2x = y - 1

→

f^{-1}(x) = 2x + 1

Example 4 — Exponential evaluation.

3^{2} = 9

;

3^{-2} = 1/9

;

3^{1/2} = \sqrt{3}

Example 5 — Not invertible without restriction.

f(x) = x^2

on all

\mathbb{R}

fails horizontal line test (

f(2) = f(-2) = 4

). Restrict to

x \geq 0

to define inverse

f^{-1}(x) = \sqrt{x}

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)

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

Symbol	Meaning
$f \circ g$	apply $g$ then $f$
$f^{-1}(y)$	input that $f$ sends to $y$
$y = a^x$	exponential base $a$

Pattern families

Easy — Compose two formulas

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.

Medium — Inverse of linear or simple rational

Swap

x

and

y

, solve for

y

. State domain of inverse as range of original. Verify

f(f^{-1}(x))=x

on valid domain.

Hard — One-to-one and invertible

Use horizontal line test or monotonicity argument. For restricted domains (e.g.

x \geq 0

), justify injectivity before inverting. Composition with inverse can simplify to identity only on matched domains.

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

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

Practice loop

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