Let’s start by saying that a relation is simply a set or collection of ordered pairs. Nothing really special about it. An ordered pair, commonly known as a point, has two components which are the x and y coordinates.

This is an example of an ordered pair.

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Main Ideas and Ways How to Write or Represent Relations

As long as the numbers come in pairs, then that becomes a relation. If you can write a bunch of points (ordered pairs) then you already know how a relation looks like. For instance, here we have a relation that has five ordered pairs. Writing this in set notation using curly braces.

Relation in set notation: 

However, aside from set notation, there are other ways to write this same relation. We can show it in a table, plot it on the xy-axis, and express it using a mapping diagram.

• Relation in table

• Relation in graph

• Relation in mapping diagram

• The domain is the set of all x or input values. We may describe it as the collection of the first values in the ordered pairs.

• The range is the set of all y or output values. We may describe it as the collection of the second values in the ordered pairs.

So then in the relation below 

our domain and range are as follows:

When listing the elements of both domain and range, get rid of duplicates and write them in increasing order.

Relations and Functions

Let’s start by saying that a relation is simply a set or collection of ordered pairs. Nothing really special about it. An ordered pair, commonly known as a point, has two components which are the x and y coordinates.

This is an example of an ordered pair.

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Relation in set notation: 

However, aside from set notation, there are other ways to write this same relation. We can show it in a table, plot it on the xy-axis, and express it using a mapping diagram.

• Relation in table

• Relation in graph

• Relation in mapping diagram

• The domain is the set of all x or input values. We may describe it as the collection of the first values in the ordered pairs.
• The range is the set of all y or output values. We may describe it as the collection of the second values in the ordered pairs.

So then in the relation below 

our domain and range are as follows:

When listing the elements of both domain and range, get rid of duplicates and write them in increasing order.

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What Makes a Relation a Function?

On the other hand, a function is actually a “special” kind of relation because it follows an extra rule. Just like a relation, a function is also a set of ordered pairs; however, every x-value must be associated to only one y-value.

Suppose we have two relations written in tables,

• A relation that is not a function

Since we have repetitions or duplicates of x-values with different y-values, then this relation ceases to be a function.

• A relation that is a function

This relation is definitely a function because every x-value is unique and is associated with only one value of y.

Relations and Functions

Let’s start by saying that a relation is simply a set or collection of ordered pairs. Nothing really special about it. An ordered pair, commonly known as a point, has two components which are the x and y coordinates.

This is an example of an ordered pair.

---

Relation in set notation: 

However, aside from set notation, there are other ways to write this same relation. We can show it in a table, plot it on the xy-axis, and express it using a mapping diagram.

• Relation in table

• Relation in graph

• Relation in mapping diagram

• The domain is the set of all x or input values. We may describe it as the collection of the first values in the ordered pairs.
• The range is the set of all y or output values. We may describe it as the collection of the second values in the ordered pairs.

So then in the relation below 

our domain and range are as follows:

When listing the elements of both domain and range, get rid of duplicates and write them in increasing order.

---

What Makes a Relation a Function?

On the other hand, a function is actually a “special” kind of relation because it follows an extra rule. Just like a relation, a function is also a set of ordered pairs; however, every x-value must be associated to only one y-value.https://tpc.googlesyndication.com/safeframe/1-0-37/html/container.htm

Suppose we have two relations written in tables,

• A relation that is not a function

Since we have repetitions or duplicates of x-values with different y-values, then this relation ceases to be a function.

• A relation that is a function

This relation is definitely a function because every x-value is unique and is associated with only one value of y.

So for a quick summary, if you see any duplicates or repetitions in the x-values, the relation is not a function. How about this example though? Is this not a function because we have repeating entries in x?

Be very careful here. Yes, we have repeating values of x but they are being associated with the same value of y. The point (1,5) shows up twice, and while the point (3,-8) is written three times. This table can be cleaned up by writing a single copy of the repeating ordered pairs.

The relation is now clearly a function!

Examples of How to Determine if a Relation is also a Function

Let’s go over a few more examples by identifying if a given relation is a function or not.

Example 1: Is the relation expressed in the mapping diagram a function?

Each element of the domain is being traced to one and only element in the range. However, it is okay for two or more values in the domain to share a common value in the range. That is, even though the elements 5 and 10 in the domain share the same value of 2 in the range, this relation is still a function.null

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Example 2: Is the relation expressed in the mapping diagram a function?

What do you think? Does each value in the domain point to a single value in the range? Absolutely! There’s nothing wrong when four elements coming from the domain are sharing a common value in the range. This is a great example of a function as well.

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Example 3: Is the relation expressed in the mapping diagram a function?

Messy? Yes! Confusing? Not really. The only thing I am after is to observe if an element in the domain is being “greedy” by wanting to be paired with more than one element in the range. The element 15 has two arrows pointing to both 7 and 9. This is a clear violation of the requirement to be a function. A function is well behaved, that is, each element in the domain must point to one element in the range. Therefore, this relation is not a function.

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Example 4: Is the relation expressed in the mapping diagram a function?

If you think example 3 was “bad”, this is “worse”. A single element in the domain is being paired with four elements in the range. Remember, if an element in the domain is being associated with more than one element in the range, the relation is automatically disqualified to be a function. Thus, this relation is absolutely not a function.

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Example 5: Is the mapping diagram a relation, or function?

Let me show you this example to highlight a very important idea about a function that is usually ignored. Your teacher may give you something like this just to check if you pay attention to the details of the definition of a function.null

So far it looks normal. But there’s a little problem. The element “2” in the domain is not being paired with any element in the range.

Here’s the deal! Every element in the domain must have some kind of correspondence to the elements in the range for it to be considered a relation, at least. Since this is not a relation, it follows that it can’t be a function.

So, the final answer is neither a relation nor a function.

Definition of One-to-One Functions

A function has many types and one of the most common functions used is the one-to-one function or injective function. Also, we will be learning here the inverse of this function.

One-to-One functions define that each element of one set say Set (A) is mapped with a unique element of another set, say, Set (B).

Or

Itcould be defined as each element of Set A has a unique element on Set B.

In brief, let us consider ‘f’ is a function whose domain is set A. The function is said to be injective if for all x and y in A,

Whenever f(x)=f(y), then x=y

And equivalently, if x ≠ y, then f(x) ≠ f(y)

Onto Function Definition (Surjective Function)

Onto function could be explained by considering two sets, Set A and Set B which consist of elements. If for every element of B there is at least one or more than one element matching with A, then the function is said to be onto function or surjective function. The term for the surjective function was introduced by Nicolas Bourbaki.

Onto Function

In the first figure, you can see that for each element of B there is a pre-image or a matching element in Set A, therefore, its an onto function. But if you see in the second figure, one element in Set B is not mapped with any element of set A, so it’s not an onto or surjective function.

Into Function :

Let f : A —-> B be a function.

There exists even a single element in B having no pre-image in A, then f is said to be an into function. 

The figure given below represents a one-one function.

Into Function – Practice Problems

Problem 1 : 

Let f : A —-> B. A, B and f are defined as 

A  =  {1, 2, 3}

B  =  {5, 6, 7, 8}

f  =  {(1, 5), (2, 8), (3, 6)}

Is f into function? Explain. 

Solution :

Write the elements of f (ordered pairs) using arrow diagram as shown below

In the above arrow diagram, all the elements of A have images in B and every element of A has a unique image.

That is, no element of A has more than one image.

So, f is a function. 

There exists an element “7” in B having no pre-image in A.  

Therefore, f is into function.

Introduction to trigonometry

Trigonometry is primarily a branch of mathematics that deals with triangles, mostly right triangles. In particular the ratios and relationships between the triangle’s sides and angles.

Trigonometry functions – introduction

There are six functions that are the core of trigonometry. There are three primary ones that you need to understand completely:

• Sine  (sin)
• Cosine  (cos)
• Tangent  (tan)

The other three are not used as often and can be derived from the three primary functions. Because they can easily be derived, calculators and spreadsheets do not usually have them.

• Secant  (sec)
• Cosecant  (csc)
• Cotangent  (cot)

All six functions have three-letter abbreviations (shown in parentheses above).

Definitions of the six functions

Consider the right triangle above. For each angle P or Q, there are six functions, each function is the ratio of two sides of the triangle. The only difference between the six functions is which pair of sides we use.

In the following table

• a is the length of the side adjacent to the angle (x) in question.
• o is the length of the side opposite the angle.
• h is the length of the Hypotenuse.

Trigonometric Functions in terms of Sine and Cosine Functions

Let’s use sine and cosine functions to determine the other trigonometric functions.

• cosec  x = 1/sin x, where x ≠ nπ
• sec x = 1/cos x  where x ≠ (2n +1)π/ 2
• tan x = sin x /cos x , where x ≠ (2n +1)π/ 2
• cot x = cos x/ sin x, where x ≠ nπ

In all the above functions, n is an integer. For all the real values of x, we already know that,

• sin² x + cos² x = 1,

This lets us know that,

• 1 + tan² x = sec² x
  1 + cot² x = cosec² x

We already know the values of trigonometric ratios for the angles of 0°, 30°, 45°,60° and 90°. We use the same values for trigonometric functions as well. The values of trigonometric functions thus are as shown in the table below:

For knowing the values of cosec x, sec x and cot x we reciprocate the values of sin x, cos x and tan x, respectively. From the above discussion, we can now calculate the values of the various trigonometric functions by using the respective trigonometric ratios, as stated in the table above.