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## PROGRAMME F10

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**PROGRAMME F10**FUNCTIONS**Programme F10: Functions**Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions**Programme F10: Functions**Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions**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 The graph of y = x3**Programme F10: Functions**Processing numbers The graph of y = x1/3**Programme F10: Functions**Processing numbers The graphs of y = x3 and y = x1/3 plotted together**Programme F10: Functions**Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions**Programme F10: Functions**Composition – ‘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**Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions**Programme F10: Functions**Trigonometric functions Rotation The tangent Period Amplitude Phase difference Inverse trigonometric functions Trigonometric equations Equations of the form acos x + bsin x = c**Programme F10: Functions**Trigonometric functions Rotation For angles greater than zero and less than /2 radians the trigonometric ratios are well defined and can be related to the rotation of the radius of a unit circle:**Programme F10: Functions**Trigonometric functions Rotation By continuing to rotate the radius of a unit circle the trigonometric ratios can extended into the trigonometric functions, valid for any angle:**Programme F10: Functions**Trigonometric functions Rotation The sine function:**Programme F10: Functions**Trigonometric functions Rotation The cosine function:**Programme F10: Functions**Trigonometric functions The tangent The tangent is the ratio of the sine to the cosine:**Programme F10: Functions**Trigonometric functions Period Any function whose output repeats itself over a regular interval is called a periodic function, theregular interval being called the period of the function. The sine and cosine functions are periodic with period 2. The tangent function is periodic with period .**Programme F10: Functions**Trigonometric functions Amplitude Every periodic function possesses an amplitude that is given as the difference between the maximum value and the average value of the output taken over a single period.**Programme F10: Functions**Trigonometric functions Phase difference The phase difference of a periodic function is the interval of the input by which the output leads or lags behind the reference function.**Programme F10: Functions**Trigonometric functions Inverse trigonometric functions If the graph of y = sin x is reflected in the line y = x a function does not result.**Programme F10: Functions**Trigonometric functions Inverse trigonometric functions Cutting off the upper and lower parts of the graph produces a single-valued function that is the inverse sine function.**Programme F10: Functions**Trigonometric functions Inverse trigonometric functions Similarly for the inverse cosine function and the inverse tangent function.**Programme F10: Functions**Trigonometric functions Trigonometric equations A simple trigonometric equation is one that involves just a single trigonometric expression: For example:**Programme F10: Functions**Trigonometric functions Trigonometric equations As another example:**Programme F10: Functions**Trigonometric functions Equations of the form acos x + bsin x = c The equation acos x + bsin x = c can be rewritten as: Remembering that multiple solutions can be found by using the graph.**Programme F10: Functions**Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions**Programme F10: Functions**Exponential and logarithmic functions Exponential functions Indicial equations**Programme F10: Functions**Exponential and logarithmic functions Exponential functions The exponential function is expressed by the equation: Where e is the exponential number 2.7182818 . . . The value of ex can be found to any level of precision desired from the series expansion:**Programme F10: Functions**Exponential and logarithmic functions Exponential functions The graphs of ex and e–x are:**Programme F10: Functions**Exponential and logarithmic functions Exponential functions The general exponential function is given by y = axwhere a > 0. Since a = elna the general exponential function can be written as:**Exponential and logarithmic functions**Exponential functions The inverse exponential function is the logarithmic function expressed by the equation:**Programme F10: Functions**Exponential and logarithmic functions Indicial equations An indicial equation is an equation where the variable appears as an index and the solution of such equations requires application of logarithms.**Programme F10: Functions**Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions**Programme F10: Functions**Odd and even functions Odd and even parts Odd and even parts of the exponential function Limits of functions The rules of limits**Programme F10: Functions**Odd and even functions Given a function f with output f (x) then, assuming f (−x) is defined: If f (−x)= f (x) the function f is called an even function If f (−x) = f (x)the function f is called an odd function**Programme F10: Functions**Odd and even functions Odd and even parts If, given f (x) where f (−x) is defined then:**Programme F10: Functions**Odd and even functions Odd and even parts of the exponential function The even part of the exponential function is: The odd part of the exponential function is: Notice:**Programme F10: Functions**Odd and even functions Limits of functions There are times when a function has no defined output for a particular value of x, say x0, but that it does have a defined value for values of x arbitrarily close to x0. For example: However, so when x is close to 1 f (x) is close to 2. it is said that: the limit of f (x) as x approaches the value x = 1 is 2**Programme F10: Functions**Odd and even functions Limits of functions The limit of f (x) as x approaches the value x = 1 is 2. Symbolically this is written as:**Programme F10: Functions**Odd and even functions The rules of limits**Programme F10: Functions**Odd and even functions The rules of limits**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 • Develop the trigonometric functions from the trigonometric ratios • Find the period, amplitude and phase of a periodic function More . . .