Functions are fundamental concepts in mathematics, linking sets of values in a unique and structured way. Understanding functions is key to exploring various mathematical ideas and real-world applications.

## Terms

**Function:**A relationship that uniquely associates each element of one set (Domain) with an element of another set (Range).**Domain:**The complete set of possible input values for the function.**Range:**The complete set of possible output values produced by the function.**Inverse Function:**A function that reverses the mapping of the original function, turning the Range back into its Domain.**Mapping:**The process of associating a value from the Domain with a value in the Range. Types of mapping include:- One-to-One: Each element in the domain is linked to a unique element in the range.
- Many-to-One: Multiple elements in the domain map to the same element in the range.
- One-to-Many: A single element in the domain maps to multiple elements in the range (Note: This is not typical in functions, but is observed in more general relations).
- Many-to-Many: Multiple elements in the domain map to multiple elements in the range (Also more typical in general relations than in functions).

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**Notations**

**Function Notation:**Functions are commonly represented as $f(x), g(x),$ etc. For example, $f(x) = 2x + 5$.**Mapping Notation:**Alternatively, functions can be written using mapping notation, such as $f: x \mapsto 2x + 5$, which explicitly shows how each element in the domain maps to an element in the range.

**Mathematical Representation and Real-World Examples**

**Mathematical Representation:**Functions can be represented in various forms, such as ordered pairs, algebraic equations, graphs, mappings or tables. Each form provides a different perspective on the function's behaviour.

**Real-World Examples:**Functions appear frequently in real-world contexts. For instance, calculating the total cost based on the number of items purchased (where the number of items is the input and the total cost is the output) or converting temperatures between Celsius and Fahrenheit.

**Determining Domain and Range**

**Domain Determination:**To find the domain of a function, consider all the possible input values. For example, the domain of a square root function excludes negative numbers as they don’t yield real number outputs.**Range Determination:**The range is determined by considering the possible outputs the function can produce. For example, the range of a quadratic function $f(x) = x^2$ is all non-negative real numbers, as squaring any real number cannot produce a negative result.

Written by: Dr Rahil Sachak-Patwa

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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.