Functions and Graphs for CAT 2026: 18 Concepts + 14 PYQs
Functions CAT 2026 questions are increasingly common in recent papers and almost completely ignored in formula-content blogs online. The reason: most aspirants treat functions as advanced theory rather than as a recognition-driven topic with 18 anchor concepts. This cheatsheet pins those 18 anchors across four blocks, walks through the eight graph transformations CAT setters reuse, and closes with 14 CAT-level PYQs across domain, range, modulus, composite, and TITA-style these questions.
The CAT 2026 functions and graphs CAT pattern leans hard on three sub-types: domain and range with restriction cues (square roots, denominators, logs), modulus functions and piecewise breakdowns, and composite or recursive evaluations. The 18 concepts below are organised so each maps to one of these recognition cues, not to a textbook chapter order.
Why CAT Tests Functions Through Recognition, Not Algebra
Functions is a relatively new high-frequency topic in CAT. Earlier papers tested it lightly through one definitional question per paper; CAT 2024 and CAT 2025 increased the count to 1 to 2 per paper, often with at least one TITA question on composite or recursive functions. The pattern is unmistakable: functions are now a CAT algebra anchor, and aspirants who skip the topic leak 2 to 4 marks every paper.
The split between aspirants who score on functions and those who freeze comes down to recognition. The 18 concepts below are structured around the visual or stem cues that trigger each one: a square root in the function flags domain restriction, a piecewise definition flags piece-by-piece evaluation, an f(f(x)) flags involutive testing.
The 18 Functions and Graphs Concepts for CAT 2026
Block 1 — Domain, Range, and Function Definition (5 concepts)
Recognition cue: any question asking for valid x values, output values, or function identification.
| # | Concept | Use case |
|---|---|---|
| 1 | Function: each x maps to one y | Definition check. |
| 2 | Domain: all valid x; Range: all output y | Basic domain-range pair. |
| 3 | Square root: radicand ≥ 0; Denominator: ≠ 0; Logarithm: argument > 0 | Three domain restrictions. |
| 4 | Range of quadratic ax2+bx+c: min/max at x = −b/2a, value c − b2/4a | Vertex-based range. |
| 5 | One-to-one (injective) vs many-to-one; Onto (surjective) vs into | Function classification. |
Block 2 — Even, Odd, Periodic and Inverse (4 concepts)
Recognition cue: questions about symmetry, period, or inverse functions.
| # | Concept | Recognition cue |
|---|---|---|
| 6 | Even: f(−x) = f(x); Odd: f(−x) = −f(x) | Symmetry classification. |
| 7 | Periodic: f(x + T) = f(x); fundamental period = smallest such T | Recurring-value pattern. |
| 8 | Inverse f−1(x): swap x and y, solve for new y | Invert one-to-one function. |
| 9 | Involutive: f(f(x)) = x; e.g., f(x) = c − x or f(x) = c/x | Self-inverse structure. |
Block 3 — Modulus and Composite Functions (4 concepts)
Recognition cue: |x| in the function or composition like f(g(x)).
| # | Concept | Use case |
|---|---|---|
| 10 | Modulus: f(x) = |x| = x for x ≥ 0, −x for x < 0 | Definition and V-shape graph. |
| 11 | Piecewise break: split at the modulus zero, write linear expressions on each side | Multi-modulus expressions. |
| 12 | Composite: (f ∘ g)(x) = f(g(x)) | Function composition. |
| 13 | Recursive: f(f(x)) = expression often implies f is linear; assume f(x) = ax + b and match | TITA pattern for hidden f. |
Block 4 — Graph Transformations (5 concepts)
Recognition cue: questions asking for a transformed graph or matching a graph to a formula.
| # | Concept | Recognition cue |
|---|---|---|
| 14 | f(x) + c: shift up by c; f(x) − c: shift down | Vertical shift. |
| 15 | f(x − h): shift right by h; f(x + h): shift left | Horizontal shift. |
| 16 | −f(x): reflect across x-axis; f(−x): reflect across y-axis | Reflections. |
| 17 | a × f(x): vertical stretch (a>1) or compress (0<a<1) | Vertical scaling. |
| 18 | |f(x)|: reflect negative parts above x-axis; f(|x|): replace x < 0 with mirror of x ≥ 0 | Modulus transformations. |
Three Functions Traps That Recur in CAT Papers
Three traps recur in CAT functions CAT 2026 questions. The first is forgetting domain restrictions when computing range. For f(x) = root(x - 2), the domain is x >= 2, which limits the range to y >= 0, not all reals. The second trap is composite-function order. (f composed with g)(x) means f(g(x)), not g(f(x)). The two are different functions in general. The third trap is the modulus piecewise error: for |x - 2| + |x - 5|, the critical points are 2 and 5, giving three pieces, not two.
14 CAT-Level Functions and Graphs Questions With Solutions
Find the domain of f(x) = root(x − 5).
Radicand ≥ 0: x − 5 ≥ 0, so x ≥ 5. Answer: x ≥ 5
Find the domain of f(x) = 1/(x2 − 4).
x2 − 4 ≠ 0, so x ≠ ±2. Domain: all reals except ±2. Answer: R − {−2, 2}
Find the range of f(x) = x2 − 4x + 7.
Vertex at x = 2, value = 4 − 8 + 7 = 3. Range: y ≥ 3. Answer: [3, ∞)
Is f(x) = x3 − x even, odd, or neither?
f(−x) = −x3 + x = −(x3 − x) = −f(x). Odd function. Answer: Odd
If f(x) = 2x + 3, find f−1(x).
Swap: y = 2x + 3, then x = 2y + 3 gives y = (x − 3)/2. So f−1(x) = (x − 3)/2. Answer: (x − 3)/2
If f(x) = 10 − x, find f(f(x)).
f(f(x)) = 10 − (10 − x) = x. Involutive. Answer: x
Graph y = |x − 3|. Where is the vertex?
V-shape with vertex at (3, 0). Answer: (3, 0)
If f(x) = x + 1 and g(x) = 2x, find f(g(3)).
g(3) = 6. f(6) = 7. Answer: 7
Evaluate |2 − 5| + |3 − 1|.
|−3| + |2| = 3 + 2 = 5. Answer: 5
If f(f(x)) = 4x + 9, find f(x), assuming f is linear.
Let f(x) = ax + b. f(f(x)) = a(ax + b) + b = a2x + ab + b. Match: a2 = 4 (so a = 2 or −2) and ab + b = 9. For a = 2: 2b + b = 9, b = 3, so f(x) = 2x + 3. Answer: 2x + 3 (one solution)
The graph of y = x2 is shifted up by 4. Find the new equation.
y = x2 + 4. Answer: y = x2 + 4
The graph of y = x2 is shifted right by 3. Find the new equation.
y = (x − 3)2. Answer: y = (x − 3)2
y = f(x) is reflected across the x-axis. The new equation is?
y = −f(x). Answer: y = −f(x)
f(x) = x + 2 for all integers. Find f(f(f(f(5)))).
f(5) = 7, f(7) = 9, f(9) = 11, f(11) = 13. Answer: 13
Wire Functions Into Your CAT 2026 Algebra Cluster
Functions is the third Algebra anchor after Quadratics and Inequalities. A diagnostic-driven plan wires this block right after Inequalities so the graph-transformation reflex compounds with the wavy curve method.
Map My Functions Concept BlockWhere Functions Fit in the CAT 2026 Algebra Cluster
Functions is the third Algebra topic after Quadratics and Inequalities. A focused 3 to 4 day block covers the 18 concepts; 6 to 8 days for working professionals. The Optima Learn CAT exam guide sequences the Algebra cluster, and the CAT 2026 waitlist details page explains how the personalised planner builds the algebra sequence.
Three Reflexes That Compress Functions Solves to Under 60 Seconds
Once 18 concepts are memorised, three reflexes separate aspirants who finish CAT these questions in 60 seconds from those who take three minutes. Reflex one: domain-cue scanning. On every function, scan for square roots, denominators, and logs immediately. Reflex two: cue-first naming. Name the question type (domain, range, modulus, composite, transformation) before computing. Reflex three: linear assumption for hidden f. If the question gives f(f(x)) and asks for f, assume f is linear (or quadratic if linear fails) and match coefficients. The CAT preparation blogs library has companion cheatsheets on Quadratics, Inequalities, and Logarithms.
Common Doubts About Functions Preparation for CAT 2026
Are TITA these questions getting harder?
Yes, slightly. CAT 2024 and CAT 2025 both included one TITA function question of moderate difficulty, usually involving composition or recursion. The fix is to drill the linear-assumption shortcut for f(f(x)) = expression questions and to memorise involutive function examples.
Should I memorise inverse functions?
Memorise the procedure (swap x and y, solve for the new y), not specific examples. The procedure handles every linear and many simple nonlinear inverses CAT can throw at you.
How tricky are recent CAT questions of this type?
Recent papers lean on composite functions and modulus piecewise problems. Both reward the cue-first and piecewise-break reflexes from this cheatsheet.
How do I revise functions one week before CAT 2026?
A one-week revision plan: day one, re-read the 18-concept cheatsheet. Day two, drill domain and range. Day three, drill modulus and composite. Day four, drill the eight graph transformations. Day five, attempt 14 mixed-block PYQs under timed conditions. Day six, review every error. Day seven, scan the cheatsheet for 15 minutes only before the exam.
Final note. Functions and graphs CAT 2026 reduces to 18 concepts across four blocks. The topic rewards cue-recognition over algebra from scratch. Drill block by block, build the three reflexes, and the CAT score predictor alongside mocks will track the lift across Algebra.
