NDA Maths · Functions

Composition and Inverse of Functions

Chain functions together (f∘g) or run one backwards (f⁻¹) — the chapter's richest source of HARD questions.

Why this matters

Twenty-eight PYQs and the chapter's difficulty hot-spot — 7 of the chapter's HARD questions live here. The staples: evaluate f at a point, compose two functions (and notice f∘g ≠ g∘f), find the constant that makes two linear functions commute (a near-annual question), and invert a rational/exponential function. Order and bijectivity are where marks are won and lost.

Concept 1 of 4

Evaluating and combining functions

Intuition

To evaluate

f

at something, substitute it everywhere

x

appears — the input can be a number, another variable, or an expression like

\sin\theta

. Sums and products of functions are done pointwise:

(f+g)(x)=f(x)+g(x)

(fg)(x)=f(x)\,g(x)

Definition

Value: $f(a)$ replaces $x$ by $a$ throughout the rule.
Substitution: $f(\sin\theta)$ replaces $x$ by $\sin\theta$ (use identities to simplify).
Pointwise sum/product: $(f\pm g)(x)=f(x)\pm g(x)$ , $(fg)(x)=f(x)g(x)$ , $\left(\tfrac fg\right)(x)=\tfrac{f(x)}{g(x)}$ where $g(x)\neq0$ .

Worked example

f(x)=x+1

and

g(x)=x^2

, find

(fg)(2)

$(fg)(2)=f(2)\,g(2)$ — a product, evaluated at $x=2$ .
$f(2)=3$ , $g(2)=4$ .
Product $=3\times4=12$ .

Answer:

(fg)(2)=12

From the bank · past-year question

Example 1FunctionsEASY

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

and

g(x)=2x-3

, then what is

(fg)(1)

equal to?

[Q96 · Apr · 2023]

$(fg)$ is a product, $(f\circ g)$ is a composition

NDA writes the product as

(fg)(x)=f(x)g(x)

and the composition as

(f\circ g)(x)=f(g(x))

. They are completely different operations — read the symbol carefully before computing.

Drill 5 more on evaluating and combining functions

Concept 2 of 4

Composition of functions

Intuition

Composition feeds one function's output into the next:

(f\circ g)(x)=f(g(x))

means 'do

g

first, then

f

'. Work inside-out, and remember the order matters —

f\circ g

and

g\circ f

are usually different.

Definition

$(f\circ g)(x)=f(g(x))$ ; the range of $g$ must sit inside the domain of $f$ . Composition is associative $(f\circ(g\circ h)=(f\circ g)\circ h)$ but not commutative $(f\circ g\neq g\circ f$ in general $)$ . Iterating $f\circ f\circ\cdots$ just repeats the substitution.

Worked example

With

f(x)=2x+1

and

g(x)=x^2

, find

(f\circ g)(x)

and

(g\circ f)(x)

$(f\circ g)(x)=f(g(x))=f(x^2)=2x^2+1$ .
$(g\circ f)(x)=g(f(x))=g(2x+1)=(2x+1)^2=4x^2+4x+1$ .
They differ — concrete proof that composition is not commutative.

Answer:

(f\circ g)(x)=2x^2+1

;

(g\circ f)(x)=4x^2+4x+1

Practice this conceptself-check

Try it yourself

f(x)=3x-2

, find

(f\circ f)(2)

From the bank · past-year question

Example 2FunctionsEASY

f(x)=4x+3

, then what is

f\circ f\circ f(-1)

equal to?

[Q68 · Apr · 2022]

Work inside-out, and keep the order

(f\circ g)(x)=f(g(x))

means

g

acts first. Students often apply

f

first or read

f\circ g

as a product. For iterated composition, peel one layer at a time — don't try to do all of

f\circ f\circ f

in one leap.

Drill 15 more on composition of functions

Concept 3 of 4

When do two linear functions commute?

Intuition

A near-annual NDA question: given two linear functions, find the constant that makes

f\circ g=g\circ f

. Both compositions are linear, so just expand both, match coefficients, and solve.

Definition

For $f(x)=ax+b$ and $g(x)=cx+d$ : $f(g(x))=acx+ad+b$ and $g(f(x))=acx+bc+d$ . The $x$ -coefficients always match (both $ac$ ), so $f\circ g=g\circ f$ reduces to the constant condition $ad+b=bc+d$ , i.e. $b(c-1)=d(a-1)$ .

Commuting condition for linear f, g

f(x)=ax+b,\ g(x)=cx+d:\quad f\circ g=g\circ f \iff b(c-1)=d(a-1)

Worked example

f(x)=2x+1

and

g(x)=3x+d

satisfy

f\circ g=g\circ f

, find

d

Here $a=2,\ b=1,\ c=3$ . Use $b(c-1)=d(a-1)$ .
$1\cdot(3-1)=d\cdot(2-1)\Rightarrow 2=d$ .

Answer:

d=2

From the bank · past-year question

Example 3FunctionsMODERATE

f(x) = 4x+1

and

g(x) = kx+2

such that

fog(x) = gof(x)

, then what is the value of k?

[Q72 · Sep · 2022]

Drill 3 more on when do two linear functions commute?

Concept 4 of 4

Inverse of a function

Intuition

The inverse

f^{-1}

undoes

f

: if

f

sends

a\mapsto b

, then

f^{-1}

sends

b\mapsto a

. It exists only when

f

is a bijection, and its graph is the mirror image of

f

across the line

y=x

Definition

To find $f^{-1}$ : write $y=f(x)$ , **solve for $x$ ** in terms of $y$ , then swap names. Domain and range swap: $\text{dom}(f^{-1})=\text{range}(f)$ . Properties: $f^{-1}\circ f=\text{id}$ , and the graph of $f^{-1}$ is the reflection of $f$ in $y=x$ . Note $f^{-1}(x)\neq\dfrac{1}{f(x)}$ .

Worked example

Find the inverse of

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

x\neq1

Set $y=\dfrac{3x+2}{x-1}$ and clear the fraction: $y(x-1)=3x+2$ .
Expand and collect $x$ : $yx-y=3x+2\Rightarrow x(y-3)=y+2$ .
Solve: $x=\dfrac{y+2}{y-3}$ ; rename $y\to x$ .

Answer:

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

Practice this conceptself-check

Try it yourself

Find the inverse of

f(x)=2x-5

From the bank · past-year question

Example 4FunctionsMODERATE

f(x)=\dfrac{x-2}{x+2}

x\neq-2

, then what is

f^{-1}(x)

equal to?

[Q86 · Apr · 2019]

Inverse needs a bijection — and $f^{-1}\neq1/f$

Only one-one onto functions have an inverse;

x^2

\mathbb{R}

has none (not one-one). And

f^{-1}

is the undo function, not the reciprocal —

f^{-1}(x)

is generally nothing like

\dfrac{1}{f(x)}

. When finding it, remember the domain of

f^{-1}

is the range of

f

Drill 3 more on inverse of a function

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

When do two linear functions commute?
Commuting condition for linear f, g
$f(x)=ax+b,\ g(x)=cx+d:\quad f\circ g=g\circ f \iff b(c-1)=d(a-1)$

Watch out for (3)

$(fg)$ is a product, $(f\circ g)$ is a composition
→ Evaluating and combining functions
Work inside-out, and keep the order
→ Composition of functions
Inverse needs a bijection — and $f^{-1}\neq1/f$
→ Inverse of a function

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1FunctionsHARD

For the following two (02) items: Consider the function

f(x)=\dfrac{x}{1-x}

(x>0,\,x\neq1)

What is

(1-x)f(\sqrt{x})+xf(\sqrt{x}+1)

equal to?

[Q77 · Sep · 2025]

Example 2FunctionsEASY

Directions for the following two (02) items : Read the following information and answer the two items that follow : Let

f(x) = x^{2}

g(x) = \tan x

and

h(x) = \ln x

What is

[f \circ (f \circ f)](2)

equal to ?

[Q80 · Sep · 2019]

Example 3FunctionsHARD

f(x)=ax-b

and

g(x)=cx+d

are such that

f(g(x))=g(f(x))

, then which one of the following holds?

[Q73 · Apr · 2024]

Example 4FunctionsMODERATE

For the following two (02) items: Consider the function

f(x)=\dfrac{10^x-10^{-x}}{10^x+10^{-x}}

What is the inverse of the function?

[Q90 · Sep · 2025]

Example 5FunctionsMODERATE

f(x)=x(4x^{2}-3)

, then what is

f(\sin\theta)

equal to?

[Q98 · Apr · 2023]

Drill every past-year question on this subtopic

28 questions from the bank — paginated, with cart and Word-export support.

Drill the 28 questions