core3 revision sheet（core3 revision sheet）.doc

Core 3 Revision Sheet Functions The domain is the input of the function The range is the output of the function Modulus functions means the ‘magnitude’ of x and is always positive. When sketching modulus functions, the graph will be reflected in the x axis. When solving modulus equations, remember to find the solutions when it is positive and negative. Example Solve = 7 x – 3 = 7 OR x – 3 = -7 x = 10 x = -4 Composite Functions When finding composite functions, apply the functions one at a time, starting with the function closest to x. Example f(x) = x2 + 3 g(x) = x + 5 Find fg(x) fg(x) = f(x + 5) = (x + 5)2 + 3 = x2 + 10x + 25 + 3 = x2 + 10x + 28 To find the range of composite functions, find the range of the first function being applied then use that as the domain of the next function being applied. This output will give the range of the overall composite function. Inverse Functions Remember that an inverse function can only be found if the original function is one-to-one. This can be achieved by restricting the domain. Example f(x) = find f(x) let y = y(x – 1) = 2x + 3 yx – y = 2x + 3 yx – 2x = 3 + y x(y – 2) = 3 + y x = Therefore f(x) = Drawing graphs of inverse functions Reflect the original graph in the line y = x You may be asked to draw inverse graphs of exponential, natural logarithms and trigonometric graphs. Make sure you are familiar with them. Differentiation Chain Rule Product Rule Quotient Rule - this is in your formula book x as a function of y = Integration Integration by substitution determine your value for u Find and rearrange it into the form dx = du Substitute du in for dx and simplify where necessary If necessary, substitute any x values in terms of u Now integrate your new function and substitute ‘x’ back in at the end if required If it is a definite integral, you must either re