Exercise 1.2 Class 12 Math Solutions - Unit 1 Graphical Representation of Functions, Punjab Board
Step-by-step solutions to Exercise 1.2, Unit 1: Graphical Representation of Functions, for Class 12 Punjab Board students. Covers algebraic vs transcendental functions, exponential function properties, graphs and asymptotes, and introductions to hyperbolic and inverse hyperbolic functions, worked out question by question.
Exercise Overview
Exercise 1.2 of Class 12 Mathematics expands upon functional analysis, guiding students into advanced transcendental curves and their distinguishing mathematical properties under the Punjab Board curriculum. The exercise begins by distinguishing algebraic functions (polynomial, rational, radical) from transcendental functions (exponential, logarithmic, trigonometric, hyperbolic). It specifically explores the exponential family (f(x) = aˣ) and (f(x) = eˣ), evaluating their domain (−∞, ∞), strictly positive range (0, ∞), injectivity, strictly increasing or decreasing nature, and horizontal asymptote (y = 0).
The second major component of this exercise introduces hyperbolic functions- sinh(x), cosh(x), tanh(x)-along with their reciprocal and inverse counterparts. Questions require students to sketch their characteristic curves, verify fundamental hyperbolic identities (such as cosh²(x) - sinh²(x) = 1), determine symmetry (even vs odd functions), and analyze asymptotic limits. Working through these problems equips Intermediate Part 2 students with essential analytical tools before encountering calculus differentiation and integration of transcendental functions in subsequent units.
For Punjab Board examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 1.2 is a regular source of high-frequency short questions, graph-sketching items, and conceptual MCQs. These step-by-step solutions clarify asymptotic behaviors, domain constraints, and hyperbolic derivations, providing students with solid exam preparation.