The Language of 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.

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.

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.

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.

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.