{"id":9973,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9973"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"notation-of-functions-domain-and-range","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/notation-of-functions-domain-and-range\/","title":{"rendered":"Notation Of Functions, Domain And Range"},"content":{"rendered":"
\n
\n

KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

\n

Unit: <\/strong>Functions<\/strong><\/h2>\n

Chapter: <\/strong>Notation of Functions<\/strong><\/h3>\n

Reference: – Definition of a Function, Function Notation, Evaluating a Function, Domain of a Function, Range of a Function, Independent and Dependent Variables, Multiple Function Notations, Piecewise Function Notation, Arithmetic with Functions, Composite Function Notation, Implicit vs. Explicit Function Notation, Function as a Mapping Rule, Inverse Function Notation<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • Definition of a Function & Function Notation<\/li>\n
  • Domain of a Function & Range of a Function<\/li>\n
  • Multiple Function Notations & Piecewise Function Notation<\/li>\n
  • Function as a Mapping Rule & Inverse Function Notation<\/li>\n<\/ul>\n
      \n
    1. Definition of a Function<\/strong>
      \n\tA function is a special type of relation in which each input from a given set (called the domain) is associated with exactly one output in another set (called the range). It ensures a unique output for every input.<\/li>\n<\/ol>\n

      \u00a0<\/p>\n

        \n
      1. Function Notation<\/strong>
        \n\tFunction notation is a symbolic way of representing functions using symbols such as f(x), where f is the name of the function and x is the input variable. It emphasizes the idea of input-output relationships.<\/li>\n<\/ol>\n

        \u00a0<\/p>\n

          \n
        1. Evaluating a Function<\/strong>
          \n\tEvaluating a function means determining the output value that corresponds to a specific input, by applying the rule defined by the function notation.<\/li>\n<\/ol>\n

          \u00a0<\/p>\n

            \n
          1. Domain of a Function<\/strong>
            \n\tThe domain of a function is the complete set of all input values for which the function rule is defined and produces a valid output.<\/li>\n<\/ol>\n

            \u00a0<\/p>\n

              \n
            1. Range of a Function<\/strong>
              \n\tThe range of a function is the complete set of all output values that result from using all the valid inputs in the domain.<\/li>\n<\/ol>\n

              \u00a0<\/p>\n

                \n
              1. Independent and Dependent Variables<\/strong>
                \n\tThe independent variable is the input value chosen freely, while the dependent variable is the output value that depends on the input, typically represented as f(x).<\/li>\n<\/ol>\n

                \u00a0<\/p>\n

                  \n
                1. Multiple Function Notations<\/strong>
                  \n\tFunctions may be represented with different letters or input variables, such as g(x), h(t), or P(n), depending on the context or nature of the relationship being modelled.<\/li>\n<\/ol>\n

                  \u00a0<\/p>\n

                    \n
                  1. Piecewise Function Notation<\/strong>
                    \n\tPiecewise functions are defined by different rules or expressions over different intervals of the domain, and function notation allows for specifying each condition clearly.<\/li>\n<\/ol>\n

                    \u00a0<\/p>\n

                      \n
                    1. Arithmetic with Functions<\/strong>
                      \n\tFunction arithmetic involves performing operations like addition, subtraction, multiplication, and division between two or more functions, and expressing the results using function notation.<\/li>\n<\/ol>\n

                      \u00a0<\/p>\n

                        \n
                      1. Composite Function Notation<\/strong>
                        \n\tComposite functions involve applying one function to the result of another function. This is represented as f(g(x)), which means that the output of g(x) becomes the input to f.<\/li>\n<\/ol>\n

                        \u00a0<\/p>\n

                          \n
                        1. Implicit vs. Explicit Function Notation<\/strong>
                          \n\tAn explicitly defined function gives the output directly in terms of the input. An implicitly defined function expresses a relationship between variables without directly solving for one variable in terms of the other.<\/li>\n<\/ol>\n

                          \u00a0<\/p>\n

                            \n
                          1. Function as a Mapping Rule<\/strong>
                            \n\tA function can be viewed as a rule that assigns to each element in the domain exactly one element in the range, often represented using notation and sometimes visualized using mapping diagrams.<\/li>\n<\/ol>\n

                            \u00a0<\/p>\n

                              \n
                            1. Inverse Function Notation<\/strong>
                              \n\tThe inverse of a function reverses the roles of input and output and is denoted, assuming the original function is one-to-one and has an inverse.<\/li>\n<\/ol>\n

                              \u00a0<\/p>\n

                                \n
                              1. Function Table Representation<\/strong>
                                \n\tA function table organizes pairs of input and output values, showing how each input is related to its corresponding output, using function notation to define the rule.<\/li>\n<\/ol>\n

                                \u00a0<\/p>\n

                                  \n
                                1. Graphical Interpretation of Function Notation<\/strong>
                                  \n\tOn a graph, f(x) represents the y-value corresponding to a specific x-value. The notation emphasizes that the vertical coordinate depends on the horizontal coordinate according to the function’s rule.<\/li>\n<\/ol>\n

                                  Example: –<\/u><\/strong><\/p>\n

                                  Let,<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  Find and simplify the expression:<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  Solution: –<\/strong><\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  <\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  So,<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  Now distribute:<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  Simplify: –<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  \u2705<\/strong> Five Conclusive Points<\/u><\/strong><\/p>\n

                                    \n
                                  1. Function Notation Clearly Represents Input-Output Relationships<\/strong>
                                    \n\tUsing symbols like f(x), function notation defines how each input is uniquely associated with an output, reinforcing the concept of a function as a rule or mapping.<\/li>\n<\/ol>\n

                                    \u00a0<\/p>\n

                                      \n
                                    1. Functions Must Have One Output for Every Input<\/strong>
                                      \n\tA fundamental property of functions is that each input in the domain corresponds to exactly one output, which distinguishes functions from general relations.<\/li>\n<\/ol>\n

                                      \u00a0<\/p>\n

                                        \n
                                      1. Notation Helps Distinguish Between Different Functions<\/strong>
                                        \n\tUsing various symbols (like f(x), g(t), h(n) allows multiple functions to be described and analysed simultaneously in a precise and organized manner.<\/li>\n<\/ol>\n

                                        \u00a0<\/p>\n

                                          \n
                                        1. Function Notation Supports Complex Operations and Transformations<\/strong>
                                          \n\tNotation allows for evaluating, combining (e.g., f+ g), composing (e.g., f(g(x)), and inverting functions, providing a foundation for more advanced algebraic and graphical work.<\/li>\n<\/ol>\n

                                          \u00a0<\/p>\n

                                            \n
                                          1. Function Notation Connects Algebraic, Numerical, and Graphical Representations<\/strong>
                                            \n\tWhether working with equations, tables of values, or graphs, function notation provides a consistent language to describe and interpret mathematical relationships across different forms.<\/li>\n<\/ol>\n

                                            \u00a0<\/p>\n

                                            <\/p>\n