Functions and Mapping – SS3 Mathematics Lesson Note (Second Term)

A function, also called a mapping, is a rule that assigns each element x in set A to a unique element y in set B. Set A is the domain and set B is the co-domain. The element y that x produces is its image, and the range is the collection of all the images of the domain elements.

Behavioural objectives

By the end of the lesson, students should be able to:

• define a function (mapping), domain, co-domain, image and range;
• state the images and the range of a mapping from a diagram;
• find the value of a function for a given value of x;
• find the domain of a mapping when the range is given;
• identify one-one and identity mappings.

What is a function?

A function is the rule which assigns an element x in set A to another unique element y in set B.

Set A is called the domain while set B is the co-domain.

What are the image and the range of a function?

• Image: the unique element in set B produced by an element in set A.
• Range: the collection of all the images of the elements of the domain.

Using the diagram above:

f(w) = g, f(x) = b, f(y) = f, f(z) = a

g, b, f and a are the images of the elements w, x, y and z respectively.

Range = {a, b, f, g}

How is a function written?

The rule which associates each element in set A with a unique element in set B is denoted by f: A → B.

Worked examples

Example 1: finding the value of a function

Given f(x) = 3x² + 2, find the values of (a) f(4) (b) f(-3) (c) f(-1/2).

Solution

f(x) = 3x² + 2

(a) f(4), that is x = 4
f(4) = 3(4²) + 2 = 3(16) + 2
= 48 + 2
= 50

(b) f(-3) = 3(-3)² + 2
= 3(9) + 2 = 27 + 2
= 29

(c) f(-1/2) = 3(-1/2)² + 2
= 3(1/4) + 2
= 3/4 + 2
= 3/4 + 8/4
= 11/4

Example 2: finding the domain from the range

Determine the domain D of the mapping g: x → 2x² – 1, if R = {1, 7, 17} is the range and g is defined on D.

Solution

g(x) = 2x² – 1, R = {1, 7, 17}

To find the domain, solve g(x) = each member of the range.

When g(x) = 1:
1 = 2x² – 1
1 + 1 = 2x²
x² = 2/2 = 1
x = 1

When g(x) = 7:
7 = 2x² – 1
7 + 1 = 2x²
8 = 2x²
x² = 4
x = 2

When g(x) = 17:
17 = 2x² – 1
17 + 1 = 2x²
18 = 2x²
x² = 9
x = 3

Domain D = {1, 2, 3}

What are the types of mapping?

One-one mapping

A mapping is one-one if different elements in the domain have different images in the co-domain. That is, if x₁ ≠ x₂, then f(x₁) ≠ f(x₂).

The mapping above is one-one.

NB: In an onto mapping, the range is the same as the co-domain.

Identity mapping

An identity mapping takes an element onto itself. If f: x → x is a mapping such that f(x) = x for all x ∈ X, it is an identity mapping.

Evaluation

1. Given f(x) = x² + 4x + 3, find the values of (a) f(2) (b) f(1/2) (c) f(-3).
2. Given that f(x) = ax + b, f(2) = 7 and f(3) = 12, find a and b.

Assignment

1. Define the domain, co-domain, image and range of a mapping.
2. Given f(x) = 3x² + 2, find f(2) and f(-1).
3. State what makes a mapping one-one, and what makes a mapping an identity mapping.