Functions of real variables solutions-Essential Mathematical Skills and Techniques, Exams of Mathematics

The document contains solutions to examples and problems from the course MAS152: Essential Mathematical Skills and Techniques, focusing on functions of real variables. It covers topics such as odd and even functions, continuity, limits, inverse functions, and hyperbolic functions (e.g., sinh, cosh, tanh). The solutions include step-by-step derivations, domain and range analysis, and proofs of identities. Additionally, it explores binomial expansions and series expansions for functions like \((1 + x)^n\). The document is practical, providing clear explanations and mathematical reasoning, making it a useful resource for students studying real analysis and foundational mathematical techniques.