• Explain the domain and range of a function. Under what circumstances would the domain be something other than all real numbers? Provide an example.

Domain: The domain of a function is the set of ‘input’ values; the function must be well defined for these input values. Range: The range of a function is the set of ‘output’ values that result after f is applied to every element of the domain.

**The domain will NOT be all real numbers when the horizontal distance from the origin is zero; Example: Consider the line of the form x=a, where a is any real number.
Here the domain is {a}, and the range is all real numbers. (x represents the horizontal distance from the origin)

• What is a function, in your own words? Give an example of a function using the variable x and explain how we evaluate a function for a given value of x.

Function: While the definition of a function is actually somewhat complex, we can summarize the basics as follows: If we were to take all possible values of x and plug them into the equation & solve for y, we will get exactly one value for each value of x

Example of a function: y = x2- 5x + 3
Evaluating a function: is simply asking what its value is for specific values of x; or what is the value of y when given the value of x. – plug in the value of x and then solve the equation for y.

...run,
all inputs are variable
3.1 The Production Function
Production function is a tool of analysis used in explaining the input-output relationship.
It describes the technical relationship between inputs and output in physical terms. In its
general form, it holds that production of a given commodity depends on certain specific
inputs. In its specific form, it presents the quantitative relationships between inputs and
outputs. A productionfunction may take the form of a schedule, a graph line or a curve,
an algebraic equation or a mathematical model. The production function represents the
technology of a firm.
An empirical production function is generally so complex to include a wide range of
inputs: land, labour, capital, raw materials, time, and technology. These variables form
the independent variables in a firm’s actual production function. A firm’s long-run
production function is of the form:
Q = f(Ld, L, K, M, T, t) (3.1.1)
where Ld = land and building; L = labour; K = capital; M = materials; T = technology;
and, t = time.
For sake of convenience, economists have reduced the number of variables used in a
production function to only two: capital (K) and labour (L). Therefore, in the analysis of
input-output relations, the production function is expressed as:
Q = f(K, L) (3.1.2)
Equation (3.1.2) represents the...

...Functions and graphing functions
Basics:
A function is a rule that changes input into output
A relation is any set of ordered pairs
A function is defined as a set of ordered pairs in which no two ordered pairs have the same element
A function must give exactly one unique output for each input
Also called a mapping or simply a map
The set of input numbers is called the domain
The set of output numbers is called the range
The set of all possible outputs is called the co-domain
The range is generally the subset of the co-domain however they can also be the same
Brackets:
A domain described as
That is, the square bracket means p is included. The rounded bracket means q is not included.
Number systems:
Composite functions:
When one function is followed by another function, the result is a composite function
Applying function after applying function is written in 3 different ways
All are pronounced ‘ after’ and mean ‘do followed by ’
Examples:
(i) Evaluate
(ii) Evaluate
(iii) Find the values for for which
(iv) Find
The number of people who visit a circus can be modelled by where represents theattendance of the circus days after it opens. The profit made by the circus can be modelled by where represents the profit in...

...understand the learning that needs to take place and together plan the tasks the will demonstrate that the competency has been met. Without the initial assessment will be difficult for both the assessor and the learner to judge what learning has taken place and then make any necessary adjustments to either complete the competency or develop the learning from where it was met
Students come to the classroom with a broad range of pre-existing knowledge, skills, beliefs, and attitudes, which influence how they attend, interpret and organize in-coming information. How they process and integrate new information will, in turn, affect how they remember, think, apply, and create new knowledge. Since new knowledge and skill is dependent on pre-existing knowledge and skill, knowing what students know and can do when they come into the classroom or before they begin a new topic of study, can help us craft instructional activities that build off of student strengths and acknowledge and address their weaknesses.
Once prior knowledge and skill is assessed, there is a range of potential responses, depending upon the type of course, the uniformity of results, and the availability and type of supplemental materials and alternatives. For example, if a majority of the class possesses misconceptions or weak understanding of a concept that you viewed as a critical prerequisite, you may decide to include covering it in class, provide a supplementary session on it,...

...1.1 Explain the function of assessment in learning and development.
Assessment is a way of finding out what learning has taken place. It enables the assessor to check what level of knowledge, skills and competency the candidate has throughout the qualification or programme. It starts with the assessor sitting down with the candidate at the beginning and creating an assessment plan for each stage of the candidate’s chosen course.
1.2 Define the key concepts and principles of assessment.
The concepts of assessment throughout the assessment process can include
* Accountability: the assessor has accountability to the learner and the organisation to ensure they are carrying out their role correctly. They may also have accountability to the learner’s employer or to an awarding organisation.
* Following the assessment strategy for your role to ensure you are carrying out your role correctly and working towards the required qualification.
* Benchmarking: This involves comparing what is accepted standard for a particular subject against the current position of the learner’s performance. If the learner doesn’t achieve the benchmark an evaluation will take place to plan for improvements.
* Evaluation of the assessment process should always take place to inform current and future practice.
* Types of assessment include initial assessment at the beginning of the course to identify the learners starting point, formative...

...1. Understand the principles and requirements of assessment
1.1Explain the function of assessment in learning and development
Assessment is carried out to ensure that learning has taken place. It measures the learner’s knowledge and skills in their learning area. Assessment encourages learners to ask questions on anything they have not fully understood, as learners know that they will have to prove their knowledge and understanding to the standards of the awarding body.
Learning and development are both connected. The learner needs guidance to understand what it is they have to learn, if they are on track and how they may improve. Assessment is essential for this to happen. There will be observation, teaching one to one to assess whether the learner has met the standards and if they are occupationally competent and to assess their current knowledge and skills.
If a training session has been delivered and no assessment has taken place then you cannot be sure that learning has taken place. If you do not assess the learner you cannot know their level of skill. Formative assessment is useful throughout the course and it gives the learner feedback which they can use to improve their future performance. It also allows the learner to build on their strengths and learn from mistakes by listening to the assessor’s feedback.
Assessment plays an important role in the education process as it keeps track of the work undertaken which can then allow for...

...filed Diameter
Diameter=x;
}
int getDiameter()
{
return Diameter;
}
-------------------------------------------------
//defining methods to set and get the field Price respectively
void setPrice(double y)
{
//setting the field Price
Price=y;
}
double getPrice()
{
return Price;
}
}
-------------------------------------------------
Program Plan:
* Create a class Pizza
* Define the variables
* Provide methods to get and set these variables
* Create another class TestPizza
* Initialize an instance of the class Pizza
* Set the fields of the instance with a value
* Output those fields with the use of getter functions
-------------------------------------------------
/***********************************************************************
The program TestPizza.java creates a class Pizza and defines methods to get and set variables in the class, creates another class TestPizza to demonstrate use of get and set methods of class Pizza
***********************************************************************/
Program:
//creating class Pizza
class Pizza
{
//declaring the fields in the class
String Toppings;
int Diameter;
double Price;
-------------------------------------------------
//defining methods to set and get the field Toppings respectively
void setToppings(String z)
{...

...Graphs and Function
What is the relation between the graphs and function and how was it applied in the real world?
Graphs are frequently used in national magazines and newspaper to present information about things such as the world’s busiest airports (O’Hare in China is first, Heathrow in London is sixth), about the advertising-dollar receivers in the United States (newspaper are first, radio is fourth) and about NCAA men’s golf team title winner (Yael is first, Houston is second). The function concept is very closely connected to graphs, and functions are the heart of mathematics.
I gathered my information from books especially algebra books and some are from the internet. I went to the library to look for some books and I borrowed some so I have many resources of information.
Many real-life relations between two quantities expressed in the form of equation are functions. To visualize these relationships, geometric figures called graphs are used. Modern technology provides us with graphing utilities needed to draw these graphs as well as enhance man’s knowledge of graphing techniques. Scientist and astronomers identify, visualize, and explore graphical patterns useful in analyzing data about the universe. Economist and businessmen draw mathematical models to find curves of best fit. Generally, the use of function and graphs is found in every scientist and...

...CHAPTER 4 : FUNCTIONS AND THEIR GRAPHS
4.1 Definition of Function
A function from one set X to another set Y is a rule that assigns each element in X to one element in Y.
4.1.1 Notation
If f denotes a function from X to Y, we write
4.1.2 Domain and range
X is known as the domain of f and Y the range of f. (Note that domain and range are sets.)
4.1.3 Object and image
If and , then x and y are known respectively as the objects and images of f. We can write
, , .
We can represent a function in its general form, that is
f(x) = y.
Example 4.1
a. Given that , find f(0), f(1) and f(2).
Example 4.2
a. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
Solution
a. Given that , find f(0), f(1) and f(2).
b. Given that , find the possible values of a such that
(a) f(a) = 4, (b) f(a) = a.
(a)
(b)
4.2 Graphs of Functions
An equation in x and y defines a function y = f(x) if for each value of x there is only one value
of y.
Example:
y = 3x +1, , .
The graph of a function in the x-y plane is the set of all points (x, y) where x is the
domain of f and y is the range of f.
Example
Figure 1 below shows the graph of a linear function, the square root...

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