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Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
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Functions
CAT'26 QUANT CHEATSHEET
Every functions formula and CAT shortcut you need for CAT 2026 — on one page.
Functions is one of those CAT topics that looks like pure theory until a question quietly turns it into a two-line trap. Most of it reduces to a few habits: read the domain restrictions, check whether a map is one-to-one, and know how the greatest-integer and fractional-part functions behave on negatives. Get those right and the rest — composition, inverses, graph shifts, and the no-calculus maxima tricks — follows quickly. This sheet gathers every formula you need for the CAT quant section, from domain and range through modulus, iterated functions and AM-GM optimisation, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-functions-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Functions: every formula you need
1Domain & Range
Domain is every valid input; range is every achievable output.
1/g(x) → g(x) ≠ 0 · √g(x) → g(x) ≥ 0 · log g(x) → g(x) > 0
Example: √(x−1) + 1/(x−3) ⇒ domain [1, ∞), x ≠ 3.
CAT Insight: 1/√x needs x > 0 strictly — never x ≥ 0.
2Types of Functions
Classified by how inputs map to outputs.
Injective: f(a) = f(b) ⇒ a = b
Surjective: Range = Codomain
Bijective: both → has inverse
Surjective: Range = Codomain
Bijective: both → has inverse
Example: 2x+3 is injective; x2 is not, since f(2) = f(−2) = 4.
CAT Hack: Horizontal-line test: if every horizontal line cuts the graph at most once, the map is one-to-one.
3Inverse Functions
Exists only for a bijection: swap x and y, then solve.
f(f−1(x)) = f−1(f(x)) = x
Example: f(x) = 3x−7 ⇒ f−1(x) = (x+7)/3; its graph is f mirrored in y = x.
4Composite Functions
Apply g first, then f; order matters, so read inside-out.
(f∘g)(x) = f(g(x)) · f(g(x)) ≠ g(f(x))
Example: f = 2x+1, g = x2 ⇒ f(g(x)) = 2x2+1, g(f(x)) = (2x+1)2.
5Even & Odd Functions
Even is symmetric about the y-axis; odd about the origin.
Even: f(−x) = f(x) · Odd: f(−x) = −f(x)
Example: x2+3 is even; x3−x is odd; x2+x is neither.
CAT Hack: Even powers give an even function, odd powers an odd one, and a mix is neither.
6Modulus Function
|x| strips the sign: the distance of x from 0.
|x| < a ⇒ −a < x < a
|x| > a ⇒ x < −a or x > a
|x| > a ⇒ x < −a or x > a
Example: |2x−5| = 3 ⇒ x = 1 or x = 4.
CAT Favourite: |x−a| + |x−b| = |a−b| holds exactly when x lies between a and b — a recurring CAT pattern.
7Greatest Integer [x]
[x] is the largest integer not exceeding x.
[x] ≤ x < [x]+1 · [x+n] = [x]+n
Example: [3.7] = 3; [−2.3] = −3; [−4] = −4.
Common Mistake: Negatives floor toward −∞: [−2.7] = −3, not −2.
8Fractional Part {x}
{x} is the decimal part of x, always in [0, 1).
{x} = x − [x] · x = [x] + {x}
Example: {3.7} = 0.7; {−2.3} = 0.7 (not −0.3).
CAT Insight: {x} is never negative, and {any integer} = 0.
9Graph Transformations
Shift, scale and reflect a known graph to sketch fast.
f(x−a) → right · f(x)+b → up
−f(x) → flip x-axis · f(−x) → flip y-axis
−f(x) → flip x-axis · f(−x) → flip y-axis
Example: y = |x2−4|: the below-axis dip flips up, so vertex (0, −4) → (0, 4).
CAT Hack: Inside change acts horizontally and in the opposite direction; outside change acts vertically and as written.
10Iterated Functions
Apply f to its own output repeatedly and look for a cycle.
fn(x) = f(fn−1(x))
if fk(x) = x, the period divides k
if fk(x) = x, the period divides k
Example: f(x) = 1/(1−x) cycles every 3, so f100(x) = f1(x) since 100 = 3×33+1.
11Functional Equations
Deduce the formula from a stated property of f.
f(x+y) = f(x)+f(y) → cx
f(x+y) = f(x)·f(y) → ax
f(xy) = f(x)+f(y) → log x
f(x+y) = f(x)·f(y) → ax
f(xy) = f(x)+f(y) → log x
Example: f(x+y) = f(x)·f(y), f(1) = 3 ⇒ f(5) = 35 = 243.
CAT Hack: Plug x = 0 and y = 0 first, then try y = x — that usually cracks the equation open.
12Maxima & Minima
Optimise without calculus: completion, AM-GM and the median trick.
Vertex at x = −b/2a · x + k/x ≥ 2√k (x > 0)
Example: min of x + 4/x for x > 0 is 4, at x = 2 (AM-GM).
CAT Insight: |x−a1| + … + |x−an| is minimum at the median of the ai.
CAT exam shortcuts, traps & revision
13
CAT Exam Shortcuts
- Domain rules: 1/g → g ≠ 0; √g → g ≥ 0; log g → g > 0
- Bijective (injective + surjective) means an inverse exists
- Composite: read inside-out; f(g(x)) ≠ g(f(x)) in general
- Even: f(−x) = f(x); Odd: f(−x) = −f(x)
- |x| < a ⇒ −a < x < a; |x| > a ⇒ x < −a or x > a
- x + k/x ≥ 2√k for x > 0 (AM-GM)
14
Most Common CAT Traps
- Flooring negatives the wrong way: [−2.7] = −3, not −2.
- Reading {−2.3} as −0.3 — the fractional part is never negative (it is 0.7).
- Assuming f(g(x)) = g(f(x)); composition order matters.
- Forgetting domain limits: 1/√x needs x > 0 strictly.
- Calling x2+x even or odd — mixed powers are neither.
15
30-Second Revision Box
- Domain: reciprocal, even root, log restrictions
- Injective / Surjective / Bijective → inverse
- (f∘g)(x) = f(g(x)); order matters
- Even / odd symmetry: y-axis / origin
- [x] and {x}: {x} = x − [x], always in [0, 1)
- Graph: inside = horizontal & opposite; outside = vertical & as written
- x + k/x ≥ 2√k (AM-GM); sum of |x−ai| minimal at the median
Functions rewards pattern-spotting over grinding — recognise whether a question is really about a domain restriction, a composition order, or a hidden AM-GM, and it collapses fast. Drill this sheet until the greatest-integer and modulus rules are reflex, then test them on full sets and track progress with the CAT score predictor. For more topic guides, browse the Optima Learn blog or explore every study guide, and work through the full CAT exam hub for section-wise strategy. When you want structured, mentor-led prep, the team at Optima Learn can map out your plan — book a free CAT 2026 call and line up your next eight weeks.
