Functions
Functions
What a function is, how to picture it and how we write it. Explained with a coffee machine analogy, plus the domain and the range.
Properties of functions
An overview of the properties of functions: domain and range, monotonicity, even and odd functions, one-to-one functions, boundedness and extrema. With examples and graphs.
Domain of a function
What the domain of a function is and how to work it out. Splitting composite functions into outer and inner functions and finding the domain in worked examples.
Inverse functions
What an inverse function is and how to find it. The definition, conditions for existence, the graph as a reflection in a line, and worked examples.
Even and odd functions
Even and odd functions: definitions, properties and graphs. How to check whether a function is even or odd by working it out, and what happens with combined functions.
Linear functions
The linear function y=mx+b and its graph, a straight line. Types of linear functions, their properties and sample graphs of each type.
Powers and roots
Powers and roots: positive, negative and fractional indices. Laws of indices, converting roots to powers and an online calculator.
Roots
What a root is and how to work with it. Square roots and higher roots, roots of negative numbers, converting to powers, formulae and a calculator.
Logarithmic functions
The logarithmic function as the inverse of the exponential function. Properties, graphs, laws of logarithms, the natural logarithm and an online calculator.
Quadratic functions
The quadratic function and its graph, a parabola. Properties, convexity, intercepts with the axes and finding the coordinates of the turning point by completing the square.
Reciprocal functions
Reciprocal functions and linear fractional functions, whose graph is a hyperbola. The function k/x, asymptotes, domain, range and converting the general form, shown in a worked example.
Exponential functions
What an exponential function is and what properties it has. The definition, conditions on the base, graphs for different values and an explanation of exponential growth.
Exponential growth
What exponential growth is and why our intuition underestimates it. Rice on a chessboard, the rule of 70, water lilies on a pond, epidemics and a folded sheet of paper.
Trigonometric functions
The trigonometric functions sine, cosine, tangent and cotangent in a right-angled triangle. Definitions, exact values and what they are used for.
