# Definition of a Function

Definition of a Function

• A function is an operation that takes an input value, performs a certain calculation on it, and produces an output value.

• Every function contains two main parts: inputs and outputs which are also known as domain and range respectively.

Important Features to Note:

• The domain comprises all possible input values for the function.

• The range is the set of all possible output values that are produced after performing the function operation.

• A function is said to be well-defined if every valid input produces a unique output.

Rules of Functions:

• A function should produce only one output for each input; this concept is also known as mapping. For example: f(x) = 3x + 1.

• When drawing a function graphically, it must pass the vertical line test. If any vertical line crosses the function at more than one point, then it is not a function.

• Function notation is written as: f(x) where f is the name of the function and x is the input to the function.

Types of Functions:

• Linear Function: It has a constant rate of change. The graph of a linear function is a straight line.

• Quadratic Function: The highest degree of a variable is 2. The graph is a parabola, for example, f(x) = x^2.

• Exponential Function: The variable acts as an exponent, for example, f(x) = 2^x.

Remember, understanding the definition and properties of a function is crucial as it forms the basis for much of the further study in mathematics and is frequently used in the AQA Further Mathematics Level 2 exam.