That is, the square bracket means p is included. The rounded bracket means q is not included. Number systems: Composite functions: When one function is followed by another function, the result is a composite function Applying function after applying function is written in 3 different ways All are pronounced ‘ after’ and mean ‘do followed by ‘ Examples: (i) Evaluate (ii) Evaluate (iii) Find the values for for which (iV) Find The number of people who visit a circus can be modelled by where represents theattendance of the circus days after it opens.The profit made by the circus can be modelled by where represents the profit in euros for he circus on a day when people attend (i) Find the number of people who were in attendance on the fourth day (ii) Find the profit made on the fourth (iii) Use these functions to find a new function that will give the profit made by the circus days after the circus opens (iv) How much profit does the circus make on the third day Graphing functions: To graph a function, find points which satisfy the function by substituting values in for x (inputs) and finding the corresponding y values (outputs).Plot these points and join them up to obtain the graph of the function Vertical line test: If any imaginary’ vertical line cuts the graph at only one point then it is a function.

If any imaginary vertical line cuts the graph at more than one point then it is not a function.Quadratic functions: A quadratic function is usually given by Two distinct real roots” Two real roots both of which are equal= No real roots= Maximum and minimum values can be found by completing the square such Can also be found using differentiation (ii) Find the coordinates of the minimum point on the graph (ii) Find the coordinates of the maximum point on the graph (iii) Sketch the graph Cubic functions:Generally in the shapes Polynomials of higher powers: As the power increases, the graph will have more turning points The maximum number of turning points is one fewer than the highest power Exponential functions: Exponential functions are given in the form (i) Graph the function (ii) From the graph estimate the point of intersection (iii) Find the range of value of x for which (iv) Evaluate Exponential function the graphs of represents exponential decay A radioactive substance is decaying. Initially 5 kg of the radioactive substance was present and four weeks later half of it remained.The mass, R kg, of the ubstance at time t weeks can be modelled by (i) State the value of (ii) Find the value of k correct to three decimal places (iii) Sketch the graph with domain (iv) Find correct to one decimal place, how long it will take for 60% of the substance to decay Injective, surjective and bijective functions: Injective is one to one mapping (range may not equal the codomain) Surjective is many to one mapping (range and codomain are equal) Bijective is perfect one to one mapping (range and codomain are equal) If a horizontal line intersects at most once.