1. “Teach A Level Maths” Vol. 2: A2 Core Modules
1: Functions

2. A function is a rule , which calculates values of
for a set of values of x.
e.g and are functions.
is called the image of x
Another Notation
means
is often replaced by y.

3. 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.

4. A bit more jargon
To define a function fully, we need to know the values of x that can be used.
The set of values of x for which the function is defined is called the domain.
In the function any value can be substituted for x, so the domain consists of
all real values of x
We write R
We say “ real ” values because there is a branch of mathematics which deals with numbers that are not real.
stands for the set of all real numbers R
means “ belongs to the set ”
So, means x belongs to the set of real numbers R

5. What is a real number?
The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as p and the 2;
or, a real number can be given by an infinite decimal representation, such as , where the digits continue in some way;
or, the real numbers may be thought of as points on an infinitely long number line.
Real numbers measure continuous quantities
such as weight, height, speed, time.

6. The range of a function is the set of values given by .
e.g. Any value of x substituted into gives a positive ( or zero ) value.
So the range of is
If , the range consists of the set of
y-values, so
domain: x-values
range: y-values
Tip: To help remember which is the domain and which the range, notice that d comes before r in the alphabet and x comes before y.

7. 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

8. e.g. 1 Sketch the function where
and write down its domain and range.
Solution: The quickest way to sketch this quadratic function is to find its vertex by differentiating.
so the vertex is

9. BUT we could have drawn the sketch for any values of x.
So, the graph of is 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 ) R
BUT there are no y-values less than -5, . . .
so the range is
( y is any real number greater than, or equal to, -5 )

10. ) ( x f y = e.g.2 Sketch the function where .
Hence find the domain and range of
Solution: is a translation from of ) ( x f y =
so the graph is:
domain: x-values
range: y-values
( We could write instead of y )

11. Asymptotes: if a function approaches a particular x or y value then the equation of this value is called an asymptote.
Notice how each value of x maps (links) to ONLY one value of y .
This is the definition of a function.
If each value of x maps to MORE than one value of y it is NOT a function

12. Vertical asymptote at x = 2
As you cannot divide by zero
x – 2  0
Domain : x  2
Vertical asymptote at x = 2
The function approaches the line x = 2 but never meets it
Horizontal asymptote at y = 3.
The graph approaches the line y = 3 but never meets it.
Range : y  3 or y<3 and y>3

13. SUMMARY
To define a function we need
a rule and a set of values.
Notation:
means
For ,
the x-values form the domain
the or y-values form the range
e.g. For ,
the domain is
the range is or R

14. Exercise
1. Sketch the functions where
For each function write down the domain and range
Solution:
(a)
(b)
domain: R
domain: R
range: R
range:

15. x + 3 must be greater than or equal to zero.
We can sometimes spot the domain and range of a function without a sketch.
e.g. For we notice that we can’t square root a negative number ( at least not if we want a real number answer ) so,
x + 3 must be greater than or equal to zero.
So, the domain is
The smallest value of is zero.
Other values are greater than zero.
So, the range is

16. One-to-one and many-to-one functions
Consider the following graphs
and
Each value of x maps to only one value of y . . .
Each value of x maps to only one value of y . . .
and each y is mapped from only one x.
BUT many other x values map to that y.

17. One-to-one and many-to-one functions
Consider the following graphs
and
is an example of a many-to-one function
is an example of a one-to-one function

18. Other one-to-one functions are:
and

19. Other many-to-one functions are:
Here the many-to-one function is two-to-one
( except at one point ! )
This is a many-to-one function even though it is one-to-one in some parts.
It’s always called many-to-one.

20. We’ve had one-to-one and many-to-one functions, so what about one-to-many?
One-to-many relationships do exist BUT, by definition, these are not functions.
e.g.
is one-to-many since it gives 2 values of y for all x values greater than 1.
This is not a function.
Functions cannot be one-to-many.
So, for a function, we are sure of the y-value for each value of x. Here we are not sure.