**Summary**

- A function is defined as a rule which assigns each element of set
*X*, one and only one element of set*Y*. i.e set and set - Domain: elements of set X
- Range: elements of set Y
- One to One Function: for each element of set
*Y*, there is a unique corresponding element in set*X*. - Many to One Function: for any element of set
*Y*, there is more than one element in set*X*. - Inverse Function:
- Composite Function: combine two functions to get a new function
- Modulus Function:

Let’s suppose we have two sets of numbers:

To define a relationship between these two sets we write a function:

or can be written as

Now we can see that for every value of set *X*, we have a value for set *Y* i.e

When

When and so on…

If the following two conditions are satisfied than the relation in called a “Function”:

i) we can say that every element of *X* has an element in set *Y*.

ii) every element of *X* has only one unique element in set *Y*.

Therefore, a function is defined as a rule which assigns each element of set *X*, one and only one element of set *Y*.

#### Domain and range

Going back to the above example which consisted of two sets:

Here elements of set *X* are known as “Domain” and elements of set *Y* are known as “Range”.

Hence we can write the domain and range as:

Domain of *f* = {10, 20, 30, 40}

Range of *f *= {32, 62, 92, 122}

*Note: we cannot define a function if its domain is not mentioned, moreover range of a function may or may not consist of all elements of the set Y.*

#### Example #1

Q. Let with domain x > 0.

Find the range of *f*.

*Solution:*

Since a function consists of a square, and we know that result of a square is always positive, therefore all values of *f(x)* will be greater than 8.

Hence the range of *f* is > 8

#### One to One Function

If we have a function *f* such that for each element of set *Y *(the range), there is a unique corresponding element in set *X* (the domain), then f is called a one to one function i.e:

given from *X* to *Y* where *X* = {4, 7, 8} is one to one as shown below:

#### Many to One Function

If we have a function *f* such that for any element of set *Y *(the range), there is more than one element in set *X* (the domain), then f is called a many to one function, i.e:

is a many to one function as the result x = 1 & -3 both correspond to 0 of set Y.

#### Composite Functions

Suppose we have two functions:

and

We can combine these two functions to get a new function , and this new function is known as a composite function, i.e:

for the above two functions means:

Since:

#### Inverse Function

Suppose we have a function , inverse of this function is the one that uses reverse relation i.e

However, remember that an inverse of a function will only exist if a function is one to one. Furthermore, if we draw the graphs of on the same axis, then the graph of will be the reflection of *f* in the line* y = x* as shown in Fig 1

#### Example #2

Q.

*Solution:*

We know:

To find inverse make *x* the subject of equation:

Hence,

**Ans**

#### Modulus Function

A modulus is defined as an absolute value of *x*. It is basically a distance of *x* from a fixed point. It is written as . A modulus function is defined as:

Therefore the graph of

is line *y = x* for *x > 0* and *y = -x* for *x < 0* as shown below in Fig 2:

#### Graphical Transformations

Suppose we have a function , some rules to remember when transforming graphs are:

- The graph of where
*c*is a constant, has the same shape of the function but is moved*c*units higher. - The graph of where
*c*is a constant, has the same shape of the function but is moved*c*units lower. - The graph of , where
*k*is a constant, has the same shape of the function but is moved*k*units to the left. - The graph of , where
*k*is a constant, has the same shape of the function but is moved*k*units to the right. - The graph of is simply a reflection of in the
*y*axis. - The graph of is a reflection of in the
*x*axis. - The graph of is stretched by a scale factor
*a*in the*y*axis. - The graph of is stretched by a scale factor in the
*x*axis.

##### Reference

- https://www.mathedup.co.uk/key-stage-5/pure-maths/core-3/modulus-function/
- https://www.s-cool.co.uk/a-level/maths/functions/revise-it/transformations