# Functions

## Functions

- A function is a mathematical rule or process which pairs each input (or x value) with exactly one output (or y-value).
- Normally, a function is expressed as an equation, such as y = f(x).
- For example, in the function y = 2x + 3, the ‘2x + 3’ is the rule that transforms any input (x-value) into an output (y-value).
- Every function should pass the vertical line test, which means that no vertical line can cross the graph more than once.
- The notation f(x) is often used, which means ‘the value of the function when the input is x’.
- If an equation is given in the format f(x) = y, then f-1(y) is the inverse function, which gives the original x value when y is put as its input.
- Functions can be linear (straight lines), quadratic (parabolas), cubic, etc. Understanding the characteristic shape of each type of function is important.
- The gradient or slope of any graph y = f(x) is given by the derivative, f’(x).
- The roots of a function are the x-values for which y = 0, i.e., the x-coordinates where the graph crosses the x-axis.
- The maximum or minimum points of a function occur when the gradient is zero, i.e., when f’(x) = 0.
- When the same value of x produces two different values of y, this does not represent a function but rather a relation.
- Functions can be composed with each other, creating composite functions. Given functions f(x) and g(x), the composite function f(g(x)) (denoted as “f of g of x”) is formed by substituting the output from g(x) into f(x).
- The domain of a function is the set of all allowable x-values. The range is the set of possible y-values.
- The transformation of functions involves the shifting, stretching, or reflecting of the function’s graph.

Using these concepts and recognising how they are applied will greatly aid in preparing for the Graphs, Functions and Calculus portion of your mathematics assessments.