1. STROUD Worked examples and exercises are in the text PROGRAMME F10 (6 th Ed) FUNCTIONS, contd (revised 29 Jan 14 – J.A.B)

2. STROUD Worked examples and exercises are in the text REMINDER All the input numbers x that a function can process are collectively called the function’s domain. So the domain of the function y = |x 1/2 | is the set of all non-negative numbers. The domain of y = 1/x is the set of all numbers except zero. [J.A.B.] The complete collection of numbers y that correspond to the numbers in the domain is called the range (or co-domain) of the function. EXERCISE [J.A.B.]: what are the ranges of the functions above? Programme F10: Functions

3. 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.

4. STROUD Worked examples and exercises are in the text Inverses of functions Programme F10: Functions The process of generating the output of a function from the input is often [J.A.B.] reversible so that what has been constructed can be de-constructed. [The textbook incorrectly implies that it is always reversible. But it isn’t if more than one input value is taken by the function to the same output value. – J.A.B.] The rule that describes the reverse process is called the inverse of the function which is labelled as f –1 as in the following [incorrect box on left – corrected in lecture—it should just show f, producing x+5 from x]:

5. STROUD Worked examples and exercises are in the text Graphs of inverses reminder 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.

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

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

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

9. 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:

10. 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.

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

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

13. STROUD Worked examples and exercises are in the text Processing numbers Composition – ‘function of a function’ Trigonometric functions Exponential and logarithmic functions (done earlier, in Graphs slides) Odd and even functions 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 Limits of functions Programme F10: Functions (but in F.11 in 7 th Ed)

15. 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 [optional] Programme F10: Functions

16. STROUD Worked examples and exercises are in the text Odd and even functions [definitions clarified by J.A.B.] Programme F10: Functions Given a function f such that f(-x) is defined (i.e., has a value) whenever f(x) is: If f (−x) = f (x) for every x in the domain of f, the function f is called an even function – If f (−x) = – f (x) for every x in the domain of f, the function f is called an odd function (notice the minus sign on the right!)

17. STROUD Worked examples and exercises are in the text Odd and even parts [definitions clarified by J.A.B.] Programme F10: Functions For f (x) such that f(-x) is defined (i.e., has a value) whenever f(x) is: and it follows that f (x) = f e (x) + f o (x)

18. STROUD Worked examples and exercises are in the text Odd and even parts of the exponential function [optional material] Programme F10: Functions The even part of the exponential function is: The odd part of the exponential function is: Notice:

19. STROUD Worked examples and exercises are in the text Limits of Functions Programme F10: Functions (but in F.11 in 7 th Ed) 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 [a rather artificial one – J.A.B.]: However, so when x is close to 1 f (x) is close to 2. [Correction by J.A.B.:] In fact, we can get f(x) to be as close as we like to 2 by making x close enough to 1. Because of this, it is said that: the limit of f (x) as x approaches 1 is 2

20. STROUD Worked examples and exercises are in the text Programme F10: Functions The limit of f (x) as x approaches 1 is 2. Symbolically this is written as: Limits of Functions The limit of 1/(ln x) as x approaches zero is zero. [J.A.B.] The limit of 1/(ln (x+2)) as x approaches -2 is zero. [J.A.B.]

21. STROUD Worked examples and exercises are in the text The rules of limits Programme F10: Functions

22. STROUD Worked examples and exercises are in the text Programme F10: Functions The rules of limits [That restriction about continuity is optional material]

23. STROUD Worked examples and exercises are in the text Programme F10: Functions The limit of 3/(x+5) as x approaches infinity or minus infinity is 0. Symbolically this is written much as above but with the positive or negative infinity symbol after the arrow under Lim. Such limits are also called asymptotic values. Limits of Functions [contd, added by J.A.B.]