1. Functions Part I Recap

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

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

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

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

6. A function must have only one y value for each x value
Is a function ?
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: , 2, 3 ….
Natural numbers N Counting numbers and zero: , 1, 2, 3
Integers Z All whole numbers: , -1, 0, 1, 2, ..
Rational numbers Q m/n where m and n are integers n≠ , 0, ½,1, 2¾
Real numbers R Rational and irrational* number , 2¾ , π, √2

8. Part II Inverse functions
What’ the inverse of f(x)=2x+4 ? X
F(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 functionx x x x
The inverse function is
An inverse function is just a rearrangement with x and y swapped.
So the graphs just swap x and y!

12. What else do you notice about the graphs?x x
is a reflection of in the line y = 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!
Solution: x

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

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

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

18. Part III Function of a function

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

35. SUMMARY
A compound function is a function of a function.
It can be written as which means
The inner function is
is not usually the same as
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. R Exercise 1. The functions f and g are defined as follows:
(a) What is the range of f ?
(b) Find (i) and (ii)
(a) The range of f is
Solution:
(b) (i)
(ii)