1. STROUD Worked examples and exercises are in the text PROGRAMME F10 FUNCTIONS

2. STROUD Worked examples and exercises are in the text Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions Programme F10: Functions

3. STROUD Worked examples and exercises are in the text Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions Programme F10: Functions

4. STROUD Worked examples and exercises are in the text 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 = x 3 The graph of y = x 1/3 The graphs of y = x 3 and y = x 1/3 plotted together Programme F10: Functions

5. STROUD Worked examples and exercises are in the text Processing numbers Programme F10: Functions 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

6. STROUD Worked examples and exercises are in the text Processing numbers Functions are rules but not all rules are functions Programme F10: 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.

7. STROUD Worked examples and exercises are in the text 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

8. STROUD Worked examples and exercises are in the text Processing numbers Functions and the arithmetic operations Programme F10: Functions Functions can be combined under the action of the arithmetic operators provided care is taken over their common domains.

9. STROUD Worked examples and exercises are in the text Processing numbers Inverses of functions Programme F10: 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:

10. STROUD Worked examples and exercises are in the text Processing numbers Graphs of inverses Programme F10: Functions 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.

11. STROUD Worked examples and exercises are in the text Processing numbers The graph of y = x 3 Programme F10: Functions

12. STROUD Worked examples and exercises are in the text Processing numbers The graph of y = x 1/3 Programme F10: Functions

13. STROUD Worked examples and exercises are in the text Processing numbers The graphs of y = x 3 and y = x 1/3 plotted together Programme F10: Functions

14. STROUD Worked examples and exercises are in the text Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions Programme F10: Functions

15. STROUD Worked examples and exercises are in the text Composition – ‘function of a function’ Programme F10: Functions 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:

16. STROUD Worked examples and exercises are in the text Composition – ‘function of a function’ Inverses of compositions Programme F10: Functions The diagram of the inverse can be drawn with the information flowing in the opposite direction.

17. STROUD Worked examples and exercises are in the text Composition – ‘function of a function’ Inverses of compositions Programme F10: Functions

18. STROUD Worked examples and exercises are in the text Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions Programme F10: Functions

19. STROUD Worked examples and exercises are in the text 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

20. STROUD Worked examples and exercises are in the text Trigonometric functions Rotation Programme F10: Functions 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:

21. STROUD Worked examples and exercises are in the text Trigonometric functions Rotation Programme F10: Functions By continuing to rotate the radius of a unit circle the trigonometric ratios can extended into the trigonometric functions, valid for any angle:

22. STROUD Worked examples and exercises are in the text Trigonometric functions Rotation Programme F10: Functions The sine function:

23. STROUD Worked examples and exercises are in the text Trigonometric functions Rotation Programme F10: Functions The cosine function:

24. STROUD Worked examples and exercises are in the text Trigonometric functions The tangent Programme F10: Functions The tangent is the ratio of the sine to the cosine:

25. STROUD Worked examples and exercises are in the text Trigonometric functions Period Programme F10: Functions Any function whose output repeats itself over a regular interval is called a periodic function, the regular 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 .

26. STROUD Worked examples and exercises are in the text Trigonometric functions Amplitude Programme F10: Functions 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.

27. STROUD Worked examples and exercises are in the text Trigonometric functions Phase difference Programme F10: Functions The phase difference of a periodic function is the interval of the input by which the output leads or lags behind the reference function.

28. STROUD Worked examples and exercises are in the text Trigonometric functions Inverse trigonometric functions Programme F10: Functions If the graph of y = sin x is reflected in the line y = x a function does not result.

29. STROUD Worked examples and exercises are in the text Trigonometric functions Inverse trigonometric functions Programme F10: Functions Cutting off the upper and lower parts of the graph produces a single-valued function that is the inverse sine function.

30. STROUD Worked examples and exercises are in the text Trigonometric functions Inverse trigonometric functions Programme F10: Functions Similarly for the inverse cosine function and the inverse tangent function.

31. STROUD Worked examples and exercises are in the text Trigonometric functions Trigonometric equations Programme F10: Functions A simple trigonometric equation is one that involves just a single trigonometric expression: For example:

32. STROUD Worked examples and exercises are in the text Trigonometric functions Trigonometric equations Programme F10: Functions As another example:

33. STROUD Worked examples and exercises are in the text Trigonometric functions Equations of the form acos x + bsin x = c Programme F10: Functions The equation acos x + bsin x = c can be rewritten as: Remembering that multiple solutions can be found by using the graph.

34. STROUD Worked examples and exercises are in the text Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions Programme F10: Functions

35. STROUD Worked examples and exercises are in the text Exponential and logarithmic functions Exponential functions Indicial equations Programme F10: Functions

36. STROUD Worked examples and exercises are in the text Exponential and logarithmic functions Exponential functions Programme F10: Functions The exponential function is expressed by the equation: Where e is the exponential number 2.7182818... The value of e x can be found to any level of precision desired from the series expansion:

37. STROUD Worked examples and exercises are in the text Exponential and logarithmic functions Exponential functions Programme F10: Functions The graphs of e x and e –x are:

38. STROUD Worked examples and exercises are in the text Exponential and logarithmic functions Exponential functions Programme F10: Functions The general exponential function is given by y = a x where a > 0. Since a = e lna the general exponential function can be written as:

39. STROUD Worked examples and exercises are in the text Exponential and logarithmic functions Exponential functions The inverse exponential function is the logarithmic function expressed by the equation:

40. STROUD Worked examples and exercises are in the text Exponential and logarithmic functions Indicial equations Programme F10: Functions An indicial equation is an equation where the variable appears as an index and the solution of such equations requires application of logarithms.

41. STROUD Worked examples and exercises are in the text Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions Odd and even functions Programme F10: Functions

42. STROUD Worked examples and exercises are in the text 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

43. STROUD Worked examples and exercises are in the text Odd and even functions Programme F10: 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

44. STROUD Worked examples and exercises are in the text Odd and even functions Odd and even parts Programme F10: Functions If, given f (x) where f (−x) is defined then:

45. STROUD Worked examples and exercises are in the text Odd and even functions Odd and even parts of the exponential function Programme F10: Functions The even part of the exponential function is: The odd part of the exponential function is: Notice:

46. STROUD Worked examples and exercises are in the text Odd and even functions Limits of functions Programme F10: Functions There are times when a function has no defined output for a particular value of x, say x 0, but that it does have a defined value for values of x arbitrarily close to x 0. 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

47. STROUD Worked examples and exercises are in the text Odd and even functions Limits of functions Programme F10: Functions The limit of f (x) as x approaches the value x = 1 is 2. Symbolically this is written as:

48. STROUD Worked examples and exercises are in the text Odd and even functions The rules of limits Programme F10: Functions

49. STROUD Worked examples and exercises are in the text Odd and even functions The rules of limits Programme F10: Functions

50. STROUD Worked examples and exercises are in the text 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...

51. STROUD Worked examples and exercises are in the text Programme F10: Functions Learning outcomes Distinguish between the inverse of a trigonometric function and the inverse trigonometric function Solve trigonometric equations using the inverse trigonometric functions Recognize that the exponential and natural logarithmic functions are mutual inverses and solve indicial and logarithmic equations Construct the hyperbolic functions from the odd and even parts of the exponential function Evaluate limits of simple functions