1. PROGRAMME F10
FUNCTIONS

2. Programme F10: Functions
Processing numbers
Composition – ‘function of a function’

3. Programme F10: Functions
Processing numbers
Composition – ‘function of a function’

4. 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

5. 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

6. 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.

7. 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.

8. 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.

9. 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:

10. 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.

11. Programme F10: Functions
Processing numbers
Graph of y = x3

12. Programme F10: Functions
Processing numbers
Graph of y = x1/3

13. Programme F10: Functions
Processing numbers
Graphs of y = x1/3 and y = x3 plotted together

14. Programme F10: Functions
Processing numbers
Composition – ‘function of a function’

15. Programme F10: Functions
Processing numbers
Composition – ‘function of a function’

16. Programme F10: Functions
Composition – ‘function of a function’
Function of a function
Inverses of compositions

17. Programme F10: Functions
Composition – ‘function of a function’
Composition – ‘function of a function’
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:

18. 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.

19. Programme F10: Functions
Composition – ‘function of a function’
Inverses of compositions

20. 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