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mathematicsprinted 48 · PDF 49

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printed 48 · PDF 49

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48 What is a function? A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers It may be thought of as a mathematical “machine” For example, if the function (rule) is “double the number and add 1”, the two mathematical operations are "multiply by 2 (×2)" and "add 1 (+1)" Putting 3 in to the function would give 2 × 3 + 1 = 7 Putting -4 in would give 2 × (-4) + 1 = -7 Putting x in would give 2x+1 The number being put into the function is often called the input The number coming out of the function is often called the output What does a function look like? A function f can be written as f(x) = … or f : x ↦ … These two different types of notation mean exactly the same thing Other letters can be used. g, h and j are common but any letter can technically be used Normally, a new letter will be used to define a new function in a question For example, the function with the rule “triple the number and subtract 4” would be written Fx=3x-4 In such cases, x would be the input and y would be the output Sometimes functions don’t have names like f and are just written as y = … eg. 3x-4 How does a function work? A function has an input x and output f(x) or y Whatever goes in the bracket (instead of x)with f, replaces the y on the other side This is the input If the input is known, the output can be calculated If the output is known, an equation can be formed and solved to find the input Composite Functions What is a composite function? A composite function is one function applied to the output of another function Composite functions may also referred to as compound functions What do composite functions look like? The notation you will see for a composite function is fg(x) This can be written as f(g(x)) and means “f applied to the output of g(x)” i.e. g(x) happens first Always apply the function on the outside to the output of the function on the inside gf(x) means g(f(x)) and means “g applied to the output of f(x)” i.e. f(x) happens first How does a composite function work?