Function Definitions and Properties

• A function is a relationship between input and output values, where each input corresponds to exactly one output.
• The set of all input values is known as the domain, while the set of output values is the range.
• The input is often represented by the variable x, and the output by the variable y.

Types of Functions

• A linear function has a constant rate of change. It is represented by the formula y = mx + c, where m is the slope, and c the y-intercept.
• A quadratic function is defined by a polynomial of second degree. It is represented by the formula y = ax² + bx + c.
• An exponential function is a function in which the variable appears in the exponent, such as y = a^x.
• The logarithmic function is the inverse of the exponential function and has the form y = loga x.

Composition of Functions

• When two functions are combined in such a way that the output of one becomes the input of the other, this is known as the composition of functions.
• If f and g are functions, the composition of f and g is written as f(g(x)) or (f o g)(x), read as 'f of g of x'.

Inverse of a Function

• The inverse of a function, denoted by f^-1(x), is a function that 'undoes' the action of the original function.
• If a function f takes a value x and transforms it into y, its inverse function f^-1 takes the value y and transforms it back to x.

Graphs of Functions

• The graph of a function is a visual representation of the relationship between input and output values.
• All points (x, y) on the graph represent valid input-output pairs, where the x-coordinate corresponds to the input value and the y-coordinate to the output value.
• Direction of graphs can be determined by viewing the left-to-right progression of a graph. If values of y increase as x increases, the function is said to be increasing. If the values of y decrease as x increases, the function is said to be decreasing.

Transformations of Functions

• A function can be transformed by changing the form of its rule or equation.
• Common transformations include translations (shifting the whole graph left, right, up, or down), stretches and compressions (altering the shape of the graph without changing its location), and reflections (flipping the graph about a line).