Relations and Functions | Class 11 Mathematics Notes

2.4.1 Some functions and their graphs

This section introduces common types of functions and their graphs. The identity function f: R → R defined by f(x) = x has domain and range as all real numbers. Its graph is a straight line passing through the origin with slope 1. The constant function f: R → R defined by f(x) = c, where c is a constant, has domain R and range {c}. Its graph is a horizontal line parallel to the x-axis. Polynomial functions are functions of the form f(x) = a0 + a1x + a2x² + ... + anx^n, where n is a non-negative integer and coefficients are real numbers. Examples include f(x) = x³ - x² + 2 and g(x) = x⁴ + √2 x. Functions like h(x) = x^(2/3) + 2x are not polynomial because of fractional exponents. The section also includes examples of quadratic and cubic functions with tables of values and their graphs. Rational functions are quotients of polynomial functions defined where the denominator is non-zero. For example, f(x) = 1/x defined on R eq {0} has a graph with two branches and vertical asymptote at x=0. The modulus function f(x) = |x| outputs the absolute value of x, with a graph shaped like a 'V'. The signum function outputs 1, 0, or -1 depending on the sign of x. The greatest integer function [x] outputs the greatest integer less than or equal to x, producing a step graph. These functions illustrate diverse behaviors and are foundational in understanding function graphs.

📊 Diagram: See figure_9: Fig 2.8 identity function graph; figure_10: Fig 2.9 constant function graph; figure_11: Fig 2.11 cubic function graph; figure_12: Fig 2.12 graph of f(x) = 1/x; figure_13: Fig 2.13 modulus function graph; figure_14: Fig 2.14 signum function graph; figure_15: Fig 2.15 greatest integer function graph.

🧪 Activity: Students complete tables of values and plot graphs for various functions to understand their shapes and properties.

🔗 Connection: This section prepares for algebraic operations on functions, combining functions to form new functions.