PROGRAMME F10 FUNCTIONS.

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Presentation transcript:

PROGRAMME F10 FUNCTIONS

Programme F10: Functions Processing numbers Composition – ‘function of a function’

Programme F10: Functions Processing numbers Composition – ‘function of a function’

Programme F10: Functions Processing numbers Functions are rules but not all rules are functions Functions and the arithmetic operations Inverses of functions Graphs of inverses The graph of y = x3 The graph of y = x1/3 The graphs of y = x3 and y = x1/3 plotted together

Programme F10: Functions Processing numbers A function is a process that accepts an input, processes the input and produces an output. If the input number is labelled x and the function is labelled f then the output can be labelled f (x) – the effect of f acting on x. Here the action of the function f is described as ^2 – raising to the power 2

Programme F10: Functions Processing numbers Functions are rules but not all rules are functions A function of a variable x is a rule that describes how a value of the variable is manipulated to generate a value of the variable y. The rule is often expressed in the form of an equation y = f (x) with the proviso that for any single input x there is just one output y – the function is said to be single valued. Different outputs are associated with different inputs. Other rules may not be single valued, for example: This rule is not a function.

Programme F10: Functions Processing numbers Functions are rules but not all rules are functions All the input numbers x that a function can process are collectively called the function’s domain. The complete collection of numbers y that correspond to the numbers in the domain is called the range (or co-domain) of the function.

Programme F10: Functions Processing numbers Functions and the arithmetic operations Functions can be combined under the action of the arithmetic operators provided care is taken over their common domains.

Programme F10: Functions Processing numbers Inverses of functions The process of generating the output of a function from the input is assumed to be reversible so that what has been constructed can be de-constructed. The rule that describes the reverse process is called the inverse of the function which is labelled:

Programme F10: Functions Processing numbers Graphs of inverses The ordered pairs of input-output numbers that are used to generate the graph of a function are reversed for the inverse function. Consequently, the graph of the inverse of a function is the shape of the graph of the original function reflected in the line f (x) = x.

Programme F10: Functions Processing numbers Graph of y = x3

Programme F10: Functions Processing numbers Graph of y = x1/3

Programme F10: Functions Processing numbers Graphs of y = x1/3 and y = x3 plotted together

Programme F10: Functions Processing numbers Composition – ‘function of a function’

Programme F10: Functions Processing numbers Composition – ‘function of a function’

Programme F10: Functions Composition – ‘function of a function’ Function of a function Inverses of compositions

Programme F10: Functions Composition – ‘function of a function’ Composition – ‘function of a function’ Function of a function Chains of functions can by built up where the output from one function forms the input to the next function in the chain. For example: The function f is composed of the two functions a and b where:

Programme F10: Functions Composition – ‘function of a function’ Inverses of compositions The diagram of the inverse can be drawn with the information flowing in the opposite direction.

Programme F10: Functions Composition – ‘function of a function’ Inverses of compositions

Programme F10: Functions Learning outcomes Identify a function as a rule and recognize rules that are not functions Determine the domain and range of a function Construct the inverse of a function and draw its graph Construct compositions of functions and de-construct them into their components