# The Language of Functions

## The Language of Functions

#### Defining Functions

- A
**function**is a special type of relation where each element from the domain (the**input**) is related to exactly one element in the codomain (the**output**). - In a function, every
**x-value**or**input**(also known as the**independent variable**) maps to a unique**y-value**or**output**(also known as the**dependent variable**). - The domain of a function contains all possible input values, while the
**range**consists of all possible output values.

#### Representation of Functions

- Functions can be represented in different ways such as
**equations**,**graphs**, or**tables**. Each of these representations provides unique insights into the behaviour of the function. - The equation form allows us to see the rule that transforms the input into the output.
- The graph gives us a visual representation of the relationship between the inputs and outputs.
- Tables explicitly present certain input-output pairs.

#### Types of Functions

**Linear functions**have graphs that are straight lines. They have a constant rate of change, defined by the slope of the line.**Quadratic functions**are characterized by an equation that can be written in the form of ax^2 + bx + c, where a ≠ 0. Their graphs are parabolas.**Exponential functions**involve the input appearing in the exponent, e.g. f(x) = a^x. They involve consistent multiplication rather than consistent addition or subtraction.**Logarithmic functions**are the inverse of exponential functions, and are used to solve for the variable in the exponent in exponential functions.**Trigonomic functions**include the sine, cosine, and tangent functions that relate to the ratios of the sides in a right triangle. They are periodic functions often used to model cyclical phenomena.

#### Operations on Functions

- Functions can be added, subtracted, multiplied, and divided to create new functions.
- The
**composite of two functions**(denoted as fog or g∘f) is the function obtained by first applying one function, then another. - The
**inverse function**of a given function f is the function that reverses the “direction” of f.

#### Transformation of Functions

- Functions can be transformed in various ways including
**translation**,**stretching**,**reflecting**, and**rotating**to form new functions. - Understanding these transformations is crucial in modelling real-world situations.