Exercise 1.3 Class 12 Math Solutions - Unit 1 Graphical Representation of Functions, Punjab Board
Step-by-step solutions to Exercise 1.3, Unit 1: Graphical Representation of Functions, for Class 12 Punjab Board students. Covers logarithmic functions and vertical asymptotes, modulus functions and V-shaped graphs, geometric construction of inverse functions, and graphical transformations (shifting and scaling), worked out question by question.
Exercise Overview
Exercise 1.3 of Class 12 Mathematics delivers an in-depth study of essential function families and graphical transformation techniques under the Punjab Board PCTB / PECTAA curriculum. The exercise begins with logarithmic functions f(x) = logₐ(x) and natural logarithms f(x) = ln(x), establishing their fundamental role as inverses of exponential functions. Students explore their domain (0, ∞), range (-∞, ∞), x-intercept at (1, 0), strict monotonicity, and vertical asymptote (x = 0).
The exercise also examines the modulus (absolute value) function f(x) = |x|, demonstrating how piecewise definitions generate characteristic V-shaped graphs with sharp corner points. Students learn geometric principles of one-to-one invertible functions, constructing the graph of an inverse function f⁻¹(x) by reflecting the original curve across the identity line (y = x). Furthermore, a comprehensive set of transformation rules is explored, training students to execute horizontal shifts f(x ± c), vertical shifts f(x ± c), vertical scaling af(x), and horizontal scaling f(ax).
For Punjab Board HSSC Part 2 examinations across all nine divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 1.3 is a key foundation for coordinate geometry and curve-sketching questions. Mastering these transformation mechanics provides students with the visual and algebraic precision needed for advanced calculus topics. These step-by-step solutions walk through each graphing problem and transformation clearly.