1. Functions Part I Recap

2. x x 11 0.. 2.1.... 1.5..... A few of the possible values of x 3.2... 2.. 11.. 33.... We can illustrate a function with a diagram The rule is sometimes called a mapping.

3. The range of a function is the set of values given by the rule. domain: x - values range: y - values The set of values of x for which the function is defined is called the domain.

4. so the range isThe graph of looks like The x -values on the part of the graph we’ve sketched go from  5 to  1... BUT we could have drawn the sketch for any values of x. ( y is any real number greater than, or equal to,  5 ) BUT there are no y -values less than  5,... x domain: So, we get ( x is any real number )

5. domain: x -values range: y -values e.g.2 Plot the function where. Hence find the domain and range of The graph looks like: ( We could write instead of y )

6. Is a function ? A function must have only one y value for each x value In other words each X gives only one F(x)

7. Notation for Domain/Range X exists for all Real numbers “R” Counting numbers Z+ Positive whole numbers: 1, 2, 3 …. Natural numbersN Counting numbers and zero: 0, 1, 2, 3 Integers Z All whole numbers:..-2, -1, 0, 1, 2,.. Rational numbers Q m/n where m and n are integers n≠0.. -1, 0, ½,1, 2¾ Real numbers R Rational and irrational* number.. -1, 2¾, π, √2

8. Part II Inverse functions What’ the inverse of f(x)=2x+4 ? X F(X) X The inverse function f -1 (x) is the reverse of f(x)

9. Calculating the Inverse In practice this is how we calculate the inverse Example: f(x)=2x+4

10. Consider the graph of the function The inverse function is

11. Consider the graph of the function The inverse function is An inverse function is just a rearrangement with x and y swapped. So the graphs just swap x and y ! x x x x

12. x x x x is a reflection of in the line y = x What else do you notice about the graphs? x The function and its inverse must meet on y = x

13. e.g.On the same axes, sketch the graph of and its inverse. N.B! x Solution:

14. e.g.On the same axes, sketch the graph of and its inverse. N.B! Solution:

15. A bit more on domain and range The domain of is. Since is found by swapping x and y, Domain Range The previous example used. the values of the domain of give the values of the range of.

16. A bit more on domain and range The previous example used. The domain of is. Since is found by swapping x and y, give the values of the domain of the values of the domain of give the values of the range of. Similarly, the values of the range of

17. N.B. The curve below does not have an inverse because it is not a function We can get around this By splitting it into two parts ( 2 y’s for 1 x)

18. Part III Function of a function

19. then, Suppose and Functions of a Function x is replaced by 3

20. and Suppose and then, Functions of a Function x is replaced by  1 x is replaced by

21. and is “a function of a function” or compound function. Suppose and then, We read as “ f of g of x ” x is “operated” on by the inner function first. is the inner function and the outer. Functions of a Function

22. Notation for a Function of a Function When we meet this notation it is a good idea to change it to the full notation. is often written as. does NOT mean multiply g by f. I’m going to write always !

23. Solution: e.g. 1 Given that and find

24. Solution:

25. e.g. 1 Given that and find

26. Solution: e.g. 1 Given that and find

27. Solution: e.g. 1 Given that and find

28. Solution: e.g. 1 Given that and find N.B. is not the same as

29. Solution: e.g. 1 Given that and find

30. Solution: e.g. 1 Given that and find

31. Solution: e.g. 1 Given that and find

32. Solution: e.g. 1 Given that and find

33. Solution: e.g. 1 Given that and find

34. Solution: e.g. 1 Given that and find

35. SUMMARY A compound function is a function of a function. It can be written as which means is not usually the same as The inner function is is read as “ f of g of x ”.

36. N.B. The order in which we find the compound function of a function and its inverse makes no difference. For all functions which have an inverse,

37. Exercise 1. The functions f and g are defined as follows: (a) The range of f is Solution: R (a) What is the range of f ? (b) Find (i) and (ii) (b) (i) (ii)