Functions

Functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping aeroplanes in the air. Functions can take input from many variables, but always give the same output, unique to that function.

Functions also allow us to visualise relationships in terms of graphs, which are much easier to read and interpret than lists of numbers.

Some examples of functions include:

• Money as a function of time. You never have more than one amount of money at any time because you can always add everything to give one total amount. By understanding how your money changes over time, you can plan to spend your money sensibly. Businesses find it very useful to plot the graph of their money over time so that they can see when they are spending too much.

• Temperature as a function of various factors. Temperature is a very complicated function because it has so many inputs, including: the time of day, the season, the amount of clouds in the sky, the strength of the wind, where you are and many more. But the important thing is that there is only one temperature output when you measure it in a specific place.

• Location as a function of time. You can never be in two places at the same time. If you were to plot the graphs of where two people are as a function of time, the place where the lines cross means that the two people meet each other at that time.

This idea is used in logistics, an area of mathematics that tries to plan where people and items are for businesses.

DEFINITION: Function

A function is a mathematical relationship between two variables, where

every input variable has one output variable.

Dependent and independent variables

In functions, the x-variable is known as the input or independent variable, because its value can be chosen freely. The calculated y-variable is known as the output or dependent variable, because its value depends on the chosen input value.

Set notation

Examples:

{x : x ∈ R, x > 0}           The set of all x-values such that x is an element
of the set of real numbers and is greater than 0.
{y : y ∈ N, 3 < y ≤ 5}     The set of all y-values such that y is a natural number,
is greater than 3 and is less than or equal to 5
{z : z ∈ Z, z ≤ 100}          The set of all z-values such that z is an integer and
is less than or equal to 100.

Interval notation

It is important to note that this notation can only be used to represent an interval of real

numbers.

Examples:

(3; 11)               Round brackets indicate that the number is not
included. This interval includes all real numbers
greater than but not equal to 3 and less than but
not equal to 11.
(−∞;−2)            Round brackets are always used for positive and
negative infinity. This interval includes all real
numbers less than, but not equal to −2.
[1; 9)                 A square bracket indicates that the number is included.
This interval includes all real numbers
greater than or equal to 1 and less than but not
equal to 9.

Function notation

This is a very useful way to express a function. Another way of writing y = 2x + 1 is f(x) = 2x + 1. We say “f of x is equal to 2x + 1”. Any letter can be used, for example, g(x), h(x), p(x), etc.

1. Determine the output value:

“Find the value of the function for x = −3” can be written as: “find f(−3)”.

Replace x with −3:

f(−3) = 2(−3) + 1 = −5  ∴ f(−3) = −5

This means that when x = −3, the value of the function is −5.

2. Determine the input value:

“Find the value of x that will give a y-value of 27” can be written as: “find x if

f(x) = 27”.

We write the following equation and solve for x:

2x + 1 = 27

∴ x = 13

This means that when x = 13 the value of the function is 27.

Representations of functions

Functions can be expressed in many different ways for different purposes.

1. Words: “The relationship between two variables is such that one is always 5 less than the other.”

2. Mapping diagram:

Input:          Function:             Output:

−3                                                    -8           

  0                    x – 5                        -5

  5                                                      0

3. Table:

Input variable (x)       −3      0      5
Output variable (y)   −8     −5      0

4. Set of ordered number pairs: (−3;−8), (0;−5), (5; 0)

5. Algebraic formula: f(x) = x − 5

Example 1  If two functions are given as f(x) = 2x + 3, and g(x) = 3 − x2, then

(a) f(2) = 2 × 2 + 3 = 7

(b) f(−3) = 2 × (−3) + 3 = −6 + 3 = −3

(c) g(0) = 3 − (0)2 = 3

(d) g(4) = 3 − (4)2 = 3 − 16 = −13

The function f(x) = 2x + 3 in example 1 may be represented as a

flow diagram.

x → multiply by 2→ 2x → add 3→ 2x+3

From the diagram it is clear that the order of the operations cannot

be confused. First multiply by 2 and then add 3.

COMPOSITE FUNCTION

The function f(x) = 2x + 3, from example 1, is composed of two

simpler functions, i.e. multiply by 2 and add 3. If these two functions

are written as h : x 7! 2x and g : x 7! x + 3 then the composition of

these two functions is written gh (sometimes as goh or g(h(x))).

Example : If h : x → 2x² and g : x →  , find the composite

function gh.

Solution

 Applying first h and then g results in the composite function

gh =

This can best be seen by using flow diagrams.

x → square x² → multiply by 2→2 x² → add 5 → 2 x²+5 → Square root→

N.B.: The composition of two functions, fg, is NOT the same as the product of two functions.

Inverse Functions

If a function f maps m to n then the inverse function, written as f−1,

maps n to m.

Example : Find the inverse of the function h : x → .

Solution

First draw a flow diagram for the function.

x → multiply by 4→ 4x → subtract 3→ 4x−3→ divide by 2 →

Now draw a flow diagram, starting from the right, with each operation

replaced by its inverse.

 → divide by 4 → 2x+3 → add 3 → 2x → multiply by 2 → x

The inverse of h : x →  is thus h⁻¹ : x →  .

Reference:

1.     Handbook on Mathematics.

2.     Book of Everything in Mathematics- Grade 9-10

3.     Wikipedia

4.     Wolfram Maths World.

Maths Education

The main goal of mathematics education is the mathematisation of the child’s thinking.

Clarity of thought and pursuing assumptions to logical conclusions is central to the mathematical enterprise. There are many ways of thinking, and the kind of thinking one learns in mathematics is an ability to handle abstractions and an approach to problem solving. Universalization of schooling has important implications for mathematics curriculum.

Mathematics being a compulsory subject of study, access to quality mathematics education is every child’s right. I want mathematics education that is affordable to every child, and at the same time, enjoyable. Mathematics education at the elementary stage should help children prepare for the challenges they face further in life.

In my vision, school mathematics takes place in a situation where: (1) Children learn to enjoy mathematics, (2) Children learn important mathematics, (3) Mathematics is a part of children’s life experience which they talk about, (4) Children pose and solve meaningful problems, (5) Children use abstractions to perceive relationships and structure, (6) Children understand the basic structure of mathematics and (7) Teachers expect to engage every child in class.

On the other hand, mathematics education in our schools is beset with problems. We identify the following core areas of concern: (a) A sense of fear and failure regarding mathematics among a majority of children, (b) A curriculum that disappoints both a talented minority as well as the non-participating majority at the same time, (c) Crude methods of assessment that encourage perception of mathematics as mechanical computation, and (d) Lack of teacher preparation and support in the teaching of mathematics. Systemic problems further aggravate the situation, in the sense that structures of social discrimination get reflected in mathematics education as well. The analysis of these problems lead us to recommend: (a) Shifting the focus of mathematics education from achieving ‘narrow’ goals to ‘higher’ goals, (b) Engaging every student with a sense of success, while at the same time offering conceptual challenges to the emerging mathematician,(c) Changing modes of assessment to examine students’ mathematization abilities rather than procedural knowledge, and (d) Enriching teachers with a variety of mathematical resources.

The shift in focus I propose is from mathematical content to mathematical learning environments, where a whole range of processes take precedence: formal problem solving, use of heuristics, estimation and approximation, optimisation, use of patterns, visualisation, representation, reasoning and proof, making connections, mathematical communication. Giving importance to these processes also helps in removing fear of mathematics from children’s minds.

I have twelve years of experience, at School level as teacher of Mathematics and in content development field. Now, I want to use the experience of mine in constructive manner by joining an educational organisation of repute as yours. I am confident that my combination of practical work experience and solid educational experience has prepared me for making an immediate contribution to your organisation. Having interned with a leading university in the public education field, I understand the level of professionalism and communication required for long-term success in this field. My background and professional approach to teaching will provide your organisation, a highly productive addition for your development program.

What I can bring is my experience of work in the field of Diagnostic assessment in Maths. Mathematics educators have always focused on assessment of student learning, carefully assessing what students have learned through chapter tests, semester tests, district tests, state tests, and norm - referenced standardized tests, just to name a few. Assessment for student learning is only effective if it is diagnostic—guiding and supporting teachers in customizing instruction for individual student needs—and provides direct and systematic interventions when the results of on going diagnostic assessments call for them.

A comprehensive system of intervention not only diagnoses problems, but also prescribes and provides the instructional materials teachers need to address students’ needs. Once students enter a unit or chapter of instruction , periodic, quick and diagnostic assessments that immediately identify learning gaps and that are linked to systematic interventions are essential to keep students on grade level; to ensure mastery of concepts, skills, and problem solving; to help students meet state and district standards; and to help students perform well on state assessments.

To ensure access to the standards for all students, classroom instruction is no longer sufficient. Effective mathematics programs now must also include a comprehensive diagnostic assessment and intervention system to keep students on grade level by supporting student mastery of skills, conceptual understanding, and problem solving. Equity in mathematics education requires nothing less. Diagnosis is an integral part of instructional decision-making. As the bridge between identification of students who may be at-risk for failure and delivery of carefully designed supplemental interventions, diagnosis provides valuable information about students’ persistent misconceptions in the targeted domain.

Making sure that there is a good “fit” between intended learning outcomes, assessment formats, and class activities/assignments is a matter of “curricular alignment.” Unless all three elements are properly aligned—outcomes, assessment, instructional format—the intended student learning outcomes, very likely, are never achieved.

That’s why the development of learning outcomes alone is insufficient, unless they are accompanied by a course design guaranteeing that these outcomes are systematically reinforced at all levels of the course.