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Mathematics 3% exam weight

Functions

Part of the CUET UG study roadmap. Mathematics topic math-016 of Mathematics.

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Functions

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision.

Functions — Key Facts for CUET • Core definition: A function f: A → B assigns exactly one output f(x) ∈ B to each input x ∈ A. The domain is A; the codomain is B; the range is {f(x) : x ∈ A}. • Most tested concept: Finding domain and range — remember to exclude values that make the denominator zero, the radicand negative, or the logarithm non-positive. • Common mistake: Confusing range with codomain — codomain is the set of all possible outputs (stated), range is the actual set reached. • Key technique: For f(g(x)), first find the domain of g, then ensure g(x) lies in the domain of f. The horizontal line test checks injectivity (one-to-one). • Important exception: The constant function f(x) = c is both even and odd only when c = 0. A function can be neither even nor odd. • Most frequent question type: Given f(x) = (expression), find its domain/range, or find f(g(x)) or f⁻¹(x) for inverse functions. ⚡ Exam tip: Before simplifying f(x), determine its domain restrictions first — otherwise you may include extraneous values that appear valid after simplification but are not.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Functions — CUET Study Guide

A function f maps every element of the domain to exactly one element in the codomain. The domain is all permissible input values; the range is the set of actual outputs. For algebraic functions, factor numerator and denominator to cancel common factors — but note that cancellation does NOT restore excluded values that were originally prohibited.

Key concepts and formulas:

  • Even function: f(−x) = f(x). Odd function: f(−x) = −f(x). Many functions are neither.
  • Periodic function: f(x + T) = f(x) for all x; smallest positive T is the period. Example: sin x has period 2π.
  • Composite function: (f ∘ g)(x) = f(g(x)). Domain of f ∘ g = {x in domain of g | g(x) in domain of f}.
  • Inverse function: f⁻¹ exists iff f is one-to-one (injective). Condition: f(a) = f(b) ⇒ a = b.
  • Algebra of functions: (f + g)(x) = f(x) + g(x); (fg)(x) = f(x) g(x); (f/g)(x) = f(x)/g(x) where g(x) ≠ 0.

Common traps:

  • Assuming f(g(x)) = g(f(x)) — composition is not commutative.
  • Forgetting that the square-root function returns the non-negative root: √(x²) = |x|, not x.
  • Missing restrictions on logarithmic arguments: log g(x) defined only when g(x) > 0.

Practice Numerical 1: Find the domain of f(x) = √(x² − 4x + 3).

  • The radicand must be ≥ 0: x² − 4x + 3 ≥ 0 → (x − 1)(x − 3) ≥ 0 → x ≤ 1 or x ≥ 3. Domain = (−∞, 1] ∪ [3, ∞).

Practice Numerical 2: If f(x) = 2x + 3 and g(x) = x² − 1, find (f ∘ g)(x) and its domain.

  • (f ∘ g)(x) = f(g(x)) = 2(x² − 1) + 3 = 2x² + 1. Domain of g is ℝ; output g(x) = x² − 1 is always ≥ −1, which is in domain of f (all real). So domain of composite is ℝ.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer timeline.

Functions — Comprehensive CUET Notes

Deeper theory and proof:

Injectivity proof using the definition: To prove f is one-to-one, assume f(a) = f(b) and deduce a = b. Example: f(x) = ax + b (a ≠ 0). If f(a) = f(b) ⇒ a + b = c ⇒ a = b, so linear functions with non-zero slope are injective.

Proving invertibility of specific functions: f(x) = eˣ is strictly increasing on ℝ, so it is injective; its inverse is f⁻¹(x) = ln x. Similarly, f(x) = x³ is injective (monotonic), with inverse f⁻¹(x) = ∛x. However, f(x) = x² is not injective on ℝ (f(2) = f(−2) = 4), so it has no inverse on ℝ. Restrict to [0, ∞) to make it invertible.

Intermediate Value Theorem (IVT) for continuity: If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then there exists at least one c ∈ (a, b) such that f(c) = 0. This is frequently used in problems about existence of roots and in proofs involving function behaviour.

Advanced classification of functions:

  • Bounded: sup |f(x)| < ∞. Example: sin x is bounded between −1 and 1.
  • Monotonic: strictly increasing (a < b ⇒ f(a) < f(b)) or decreasing. Example: f(x) = 1/x is decreasing on (0, ∞) but not on (−∞, 0).
  • Periodic with fundamental period: smallest positive T satisfying f(x + T) = f(x). Example: sin(2x) has period π, not 2π.

Limit and continuity edge cases: The function f(x) = (x² − 4)/(x − 2) simplifies to x + 2 for x ≠ 2, but at x = 2 the original expression is undefined (hole). The limit as x → 2 is 4, but f(2) is not defined. Always check domain before simplifying.

Challenging solved example: Let f(x) = { x² for x < 0; 2x + 1 for x ≥ 0 }. Find f⁻¹ if it exists.

  • f is not one-to-one on ℝ because it is not monotonic across 0: For x < 0, f(x) = x² ≥ 0, so f(−1) = 1 and f(1) = 3 from the other branch. However, on [0, ∞), f(x) = 2x+1 is strictly increasing and maps [0, ∞) to [1, ∞). On (−∞, 0), f(x) = x² maps (−∞, 0) to (0, ∞). The two ranges overlap (0, 1), so f is not injective on ℝ and has no inverse globally. If restricted to [0, ∞), f⁻¹(y) = (y − 1)/2 for y ≥ 1.