NDA Maths · Functions

Functional Equations

You are given a relation the function must satisfy — substitute clever values to pin down f(x) or a specific value.

Why this matters

Eighteen PYQs, mostly MODERATE — the chapter's 'solve for the unknown function' genre. Three patterns cover almost all of them: substitute a second argument (x → 1/x or x → 1−x) and solve the resulting linear system; recognise multiplicative/additive forms (f(xy)=f(x)f(y), f(x+y)=f(x)f(y)); or undo an argument shift like f(x+1)=… to recover f(x). Substitution is the master tool.

Concept 1 of 3

Solving by substitution (x → 1/x, x → 1−x)

Intuition

When one equation mixes

f(x)

with

f(\text{something})

, substitute that 'something' for

x

to get a second equation. Two equations, two unknowns

f(x)

and

f(\text{other})

— solve like simultaneous equations.

Definition

If a relation links $f(x)$ and $f(g(x))$ where $g(g(x))=x$ (e.g. $g(x)=\tfrac1x$ or $1-x$ ), replace $x$ by $g(x)$ to obtain a second relation, then eliminate $f(g(x))$ algebraically to isolate $f(x)$ .

Worked example

f(x)+2f(1-x)=x

for all

x

, find

f(x)

Replace $x$ by $1-x$ : $f(1-x)+2f(x)=1-x$ . $\;$ (2)
Original: $f(x)+2f(1-x)=x$ . $\;$ (1)
Compute $2\times(2)-(1)$ : $4f(x)+2f(1-x)-f(x)-2f(1-x)=2(1-x)-x\Rightarrow 3f(x)=2-3x$ .
Divide by 3.

Answer:

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

Practice this conceptself-check

Try it yourself

2f(x)+f(1-x)=x

for all

x

, find

f(x)

From the bank · past-year question

Example 1FunctionsMODERATE

4f(x)-f\left(\frac{1}{x}\right)=\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right)

, then what is

f(2)

equal to?

[Q67 · Apr · 2022]

One equation, two unknowns — make a second

You cannot read off

f(x)

from a single relation that also contains

f(1/x)

f(1-x)

. Generate the partner equation by substituting, then solve the

2\times2

system. Substituting a value that is its own partner (like

x=\tfrac12

for

x\to1-x

) can shortcut a single requested value.

Drill 4 more on solving by substitution (x → 1/x, x → 1−x)

Concept 2 of 3

Multiplicative and additive forms

Intuition

A few functional equations have signature solutions. Recognise the form and the function type follows: products go to powers, sums-to-products go to exponentials, sums-to-sums go to linear.

Definition

$f(xy)=f(x)f(y)$ : power-type, $f(x)=x^k$ ; useful values come from $f(1)=1$ and $f(1/a)=1/f(a)$ .
$f(x+y)=f(x)f(y)$ : exponential, $f(x)=a^x$ ; so $f(x)f(y)f(z)=f(x+y+z)$ .
$f(x+y)=f(x)+f(y)$ : additive (Cauchy), $f(x)=cx$ .

Worked example

f(x+y)=f(x)f(y)

for all

x,y

and

f(1)=3

, find

f(3)

The form $f(x+y)=f(x)f(y)$ means $f$ is exponential: $f(x)=f(1)^x$ .
So $f(3)=f(1)^3=3^3$ .

Answer:

f(3)=27

Practice this conceptself-check

Try it yourself

f(xy)=f(x)f(y)

and

f(3)=9

, find

f\!\left(\tfrac13\right)

From the bank · past-year question

Example 2FunctionsEASY

Let

f(x)f(y)=f(xy)

for all real

x,y

. If

f(2)=4

, then what is the value of

f\!\left(\dfrac{1}{2}\right)

[Q80 · Sep · 2024]

Drill 10 more on multiplicative and additive forms

Concept 3 of 3

Undoing an argument shift

Intuition

When the rule gives

f(\text{shifted } x)

instead of

f(x)

, introduce a new variable for the shifted argument, solve for the original

x

, and substitute back to read

f

of a bare variable.

Definition

Given $f(g(x))=h(x)$ , set $t=g(x)$ , solve $x$ in terms of $t$ , and substitute: $f(t)=h(\,x(t)\,)$ . Renaming $t\to x$ gives the explicit rule. (E.g. from $f(x+1)$ put $t=x+1\Rightarrow x=t-1$ .)

Worked example

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

, find

f(x)

Let $t=x-1\Rightarrow x=t+1$ .
Substitute: $f(t)=(t+1)^2+1=t^2+2t+2$ .
Rename $t\to x$ .

Answer:

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

From the bank · past-year question

Example 3FunctionsEASY

f(x+1) = x^2 - 3x + 2

, then what is

f(x)

equal to?

[Q51 · Apr · 2021]

Solve for the original variable before substituting

From

f(x+1)=x^2-3x+2

you must write

x=t-1

(where

t=x+1

) and plug that in — not simply replace

x

x

in the right side. Getting the shift direction backwards is the usual slip.

Drill 1 more on undoing an argument shift

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.

Watch out for (2)

One equation, two unknowns — make a second
→ Solving by substitution (x → 1/x, x → 1−x)
Solve for the original variable before substituting
→ Undoing an argument shift

Mastery check — 5 interleaved questions

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

Example 1FunctionsMODERATE

2f(x)+f(1-x)=x

, then what is

f(x)

equal to?

[Q80 · Apr · 2026]

Example 2FunctionsMODERATE

A function satisfies

f(x-y)=f(x)f(y)

, where

f(y)\neq0

. If

f(1)=0.5

, then what is

f(2)+f(3)+f(4)+f(5)+f(6)

equal to?

[Q25 · Sep · 2023]

Example 3FunctionsMODERATE

f(2x)=4x^2+1

, then for how many real values of

x

will

f(2x)

be the GM of

f(x)

and

f(4x)

[Q76 · Sep · 2024]

Example 4FunctionsHARD

Let

3f(x)+f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+1

What is

f(x)

equal to?

[Q99 · Apr · 2024]

Example 5FunctionsMODERATE

Consider the following for the items that follow: Let

f(x)

be a function satisfying

f(x+y) = f(x)f(y)

for all

x, y \in N

such that

f(1) = 2

\sum_{x=2}^{n} f(x) = 2044

, then what is the value of

n

[Q31 · Sep · 2022]

Drill every past-year question on this subtopic

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

Drill the 18 questions