1. CIE Centre A-level Pure Maths
P1 Chapter 3
CIE Centre A-level Pure Maths
© Adam Gibson

2. Domain and Range We take one number, x and make another number, f(x)
e.g.
Suppose
What is f(7)?
The complete definition of a function
must include a domain:

3. What is and is not a function
Consider the following two equations
Which one is a function?
and
One x, one y
One x, two y

4. The range is the set of
possible values for f(x)
What is the range of f(x) = +√x?
range
domain

5. COMPOSITE functions g f
Recall from page 197, that a function is a MAPPING from x to y
To make a composite function, we first apply a function f to x, and then another function g. We call this the composite function f ( g ( x ) )
Example: g f

6. ! IMPORTANT COMPOSITE functions 2 -15 -1 1/898 4 27 -15/2
Check your understanding.
Find the value of these expressions:
! IMPORTANT
-15 -1
1/898 4 27
-15/2
Remember that:
means do g first, then f

7. Review – functions - basics
Solve as many questions as possible from Exercises 9.1 and 9.2 p
Hand in at least 20 completed solutions to me.

8. INVERSE functions
Refer to page , and the box at the bottom of page 213.
You need to remember three things:.
Only a certain type of function has an inverse
There is a standard process to find an inverse which you need to learn
The graph of a function and its inverse have a very important pattern or relationship

9. INVERSE functions 2 Only a certain type of function has an inverse
Are f and g one-one?
ANSWER: g is not one-one because you cannot choose ONE x for each y

10. A function has an inverse if and only if it is one-one.
INVERSE functions 3
Can we modify g to make it one-one?
ANSWER: Yes. We just change the domain to be one side of the vertex.
Now each value of y has only one value of x.
A function has an inverse if and only if it is one-one.

11. INVERSE functions 4
There is a standard process to find an inverse which you need to learn
If y = f (x), then we write the inverse as x = f -1 ( y )
Study this Example:
Apply the method:
“make x the subject”

12. INVERSE functions 5
ANSWERS:

13. INVERSE functions 6
The graph of a function and its inverse have a very important pattern or relationship
The graph of a function and its inverse are symmetrical about the line y=x

14. INVERSE functions 7
Here is another example of inverse function symmetry. This one is very important!
“Complete the square”
Check the domain and range

15. INVERSE functions 8
Here is the process in full for a quadratic function:
Why?
one-one!
Complete the square
Make x the subject
Choose the root (+ or -) – domain or range?

17. Odd and even An odd function has An even function has
Very useful for sketching graphs!

19. Quadratic functions … are functions of the form:
How many roots does this function have?
Tasks
Write down the general solution to the equation f(x)=0
Explain how knowing the values of a, b and c can help
you to sketch the graph of f(x)
Sketch the graph of

20. Another example: the modulus function |x|
Defined as:
What is the domain of |x|?
What is the range of |x|?
Is |x| odd or even? y x

21. Finding the range Sketch a graph of each of these functions, and then
find their RANGE:
(It’s very important that you learn how to do cases like k(x) )