Functions, Graphs & Logarithms

medium importance25 formulas⚡ 15 shortcuts5 subtopics

Overview Learn (5)Formulas (25)Examples (5)Practice (71)

Function notation, graphs, modulus and logarithms — the algebra engine room of management exams. Banking and SSC papers meet it mainly in Tier 2 and higher-level sets, where 1–2 clean, formula-driven questions reward anyone who knows the standard moves.

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Learn — 5 subtopics

One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.

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Formula sheet

Every formula on one printable page, grouped by subtopic.

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Solved examples

5 exam-level questions worked step by step.

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Practice & test

71 questions — untimed practice or a timed test with analysis.

Track record in the exam

Test difficulty mix (71 questions)

22 easy35 medium14 hard

Question patterns exams keep repeating

Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.

Function value from a rule

very common

A rule like $f(x) = x^2 + 2x$ is given and a value such as $f(3)$ is asked.

Substitute the number into the rule.

Simplify in one careful line.

Example

If $f(x) = x^2 + 2x$ , find $f(3)$ .

$f(3) = 9 + 6 =$ 15

Learn this in “Functions, domain & range” →

Domain of a rational or root function

very common

'For what values of $x$ is $f$ defined?' — the rule has a denominator or a root.

Reject inputs that make the denominator zero.

Keep the inside of every root non-negative.

Example

Find the domain of $f(x) = \dfrac{x+1}{x^2 - 5x + 6}$ .

$x^2 - 5x + 6 = (x-2)(x-3)$

Reject $x = 2$ and $x = 3$ : domain is all reals except 2 and 3

Learn this in “Functions, domain & range” →

Composite function at a point

very common

Two rules $f$ and $g$ are given; $f(g(a))$ is asked for a number.

Evaluate the inner rule first.

Feed the result into the outer rule.

Example

If $f(x) = 2x + 3$ and $g(x) = x^2$ , find $f(g(2))$ .

$g(2) = 4$

$f(4) =$ 11

Learn this in “Composite & inverse functions” →

Inverse of a fractional linear function

common

$f(x) = \dfrac{ax+b}{cx+d}$ is given and $f^{-1}(x)$ or $f^{-1}(t)$ is asked.

Use $f^{-1}(x) = \dfrac{b-dx}{cx-a}$ , or swap and solve.

For $f^{-1}(t)$ with $d = -a$ , just compute $f(t)$ .

Example

Find $f^{-1}(x)$ for $f(x) = \dfrac{2x+3}{x-1}$ .

$y(x-1) = 2x + 3$ , so $x(y-2) = y+3$

$f^{-1}(x) =$ $\dfrac{x+3}{x-2}$ (check: $f(8/3) = 5$ )

Learn this in “Composite & inverse functions” →

Functional equation with small values

occasional

$f(x+y) = f(x) + f(y)$ style with one value like $f(1)$ given.

Climb one unit step at a time from the given value.

Sum type adds; product type multiplies.

Example

If $f(x+y) = f(x) + f(y)$ and $f(1) = 4$ , find $f(5)$ .

Each step adds $4$ : $f(5) = 5 \times 4 =$ 20

Learn this in “Composite & inverse functions” →

Maximum or minimum of a quadratic via the vertex

very common

'Find the maximum/minimum value of $ax^2 + bx + c$ ' — or the input where it occurs.

Vertex at $x = -b/2a$ .

$a > 0$ gives a minimum, $a < 0$ a maximum.

Substitute back for the value.

Example

Find the maximum value of $f(x) = -2x^2 + 12x - 11$ .

$x = -12/(-4) = 3$

$f(3) = -18 + 36 - 11 =$ 7

Learn this in “Graphs & transformations” →

Logarithm evaluation with the laws

very common

A log value is asked directly, or logs of products and powers must be combined.

Convert to power language.

Split products into sums; pop powers out front.

Example

Find $\log_{10} 25 + \log_{10} 4$ .

$= \log_{10}(25 \times 4) = \log_{10} 100 =$ 2

Learn this in “Logarithms” →

Log equation with a rejected root

common

An equation like $\log x + \log(x-3) = 1$ — the quadratic gives two roots.

Combine into one log, convert to power form, solve.

Reject roots with non-positive arguments.

Example

Solve $\log_{10} x + \log_{10}(x - 3) = 1$ .

$x(x-3) = 10$ , so $x = 5$ or $x = -2$

$x = -2$ breaks the domain, so x = 5

Learn this in “Logarithms” →

Modulus equation with two cases

very common

$|2x - 3| = 5$ style — a modulus equals a number.

Split into $+$ and $-$ cases (or use distance).

Both solutions usually count.

Example

Solve $|2x - 3| = 5$ .

$2x - 3 = 5$ gives $4$ ; $2x - 3 = -5$ gives $-1$

x = 4 or x = -1

Learn this in “Modulus & equations” →

Exponential equation by matching bases

common

$3^{2x-1} = 81$ style — powers on both sides.

Write both sides with the same base.

Equate exponents and solve.

Example

Solve $3^{2x-1} = 81$ .

$81 = 3^4$ , so $2x - 1 = 4$

x = 5/2

Learn this in “Modulus & equations” →

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