Understanding Functions

• A function is a special type of relation where each element of the input set (the domain) is matched with exactly one element from the output set (the range).

• The domain of a function is the set of all possible input values (usually represented by 'x') that the function can handle.

• The range of a function is the set of all possible output values (usually represented by 'y'), given the set of input values from the domain.

• In maths, functions are commonly written in the form y = f(x) where y is the output, x is the input and f is the function.

Describing Functions

• A linear function produces a straight line when graphed. The general form of a linear function is f(x) = mx + c, where m is the gradient and c is the y-intercept.

• A quadratic function produces a curve when graphed and is generally written in the form f(x) = ax^2 + bx + c.

• An exponential function has the general form f(x) = a^x, and its output increases or decreases rapidly as x increases.

• A logarithmic function is the inverse of an exponential function, typically written as f(x) = loga(x), where a is the base of the log.

Function Notation and Transformations

• The notation f(x + h) means shifting the graph of the function f(x) h units to the left. If h is negative, the graph moves to the right.

• The notation f(x) + k means shifting the graph of the function f(x) k units up. If k is negative, the graph moves down.

• If 'a' is a constant and 'k' is a function, the notation a * f(x) implies stretching the function vertically by a factor of a.

• Composite functions can be formed by applying one function to the output of another function. This is denoted as (f◦g)(x) = f(g(x)).

The Inverse Function

• An inverse function 'undoes' the work of a function. If 'f' is a function that takes x to y, then the inverse function takes y back to x.

• The notation f^-1(x) represents the inverse function.

• The graph of an inverse function is a reflection of the original function across the line y=x.

Practising transforming and drawing functions is key to solidifying your understanding of this topic. Being able to quickly identify basic function types and their inverses is important for tackling more complex problems.